Coherent structures in the wake of a SAE squareback vehicle model
CCoherent structures in the wake of a SAEsquareback vehicle model
Benjamin Bock ∗ Tplus Engineering GmbH,Steinbeisstraße 25,70771 Leinfelden-Echterdingen, Germany
The wake of a SAE squareback vehicle model is studied both experimentallyand numerically for a Reynolds-number of Re h = 1 . .The investigation focuseson the coherent structures of the intermediate to largest length and time scales.Flow field as well as base pressure fields are observed for the understanding of therelation between the signals of these quantities. Generalizations and differentiationsare made by comparison with the documented behavior of Ahmed or similar vehiclemodels or three-dimensional bluff bodies. In comparison the vortex shedding actssimilar but is restricted to the upper half of the wake of the SAE vehicle model.Due to the localization and phase behavior of the vortex shedding the connectionbetween the base pressure signals and the flow field is weak. However, the pressuresignals may be a viable feedback sensor under certain conditions, for example in flowcontrol applications. A flapping of the near wake is identified for the fluctuationsof the low frequency time scales. Key words:
Bluff body, Coherent Structures, SAE squareback model
In recent decades, the manipulation of the flow in the wake of bluff bodies has been a topicof great interest in numerous investigations. Many examples can be found in the shape opti-mization of vehicles (see comprehensive work of Hucho and Sovran, 1993) and buildings (seecomprehensive work of Xie, 2014), as well as in approaches to control these and similar basicbluff body flows in a passive or active manner (Choi et al., 2008; Kim et al., 2008; Pastooret al., 2008; Wassen et al., 2010; Krentel et al., 2010; Barros et al., 2015; Schmidt et al., 2015;Wieser et al., 2015; Littlewood and Passmore, 2012). The main objective of these efforts is tocontrol the forces of the bluff body in a desired direction or way. This means in particular to ∗ Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] A ug B.Bock reduce drag, to increase or decrease lift or to control the distribution or the dynamics of forces.The latter is relevant, for example, in the vibration of buildings (e.g. Hayashida and Iwasa,1990; Xu et al., 1992) and in the steering of a vehicle (Stoll et al., 2015). For the manipulationof such flows around bluff bodies the understanding of the dynamics and the motion of the flowin the wake is essential.In various ways the literature (see Pastoor et al., 2008; Barros et al., 2015; Bock et al.,2016) demonstrates that fluctuations in the wake flow caused by vortex motions or coherentstructures are sources of losses which contribute to aerodynamic drag. Hence the reduction orthe prevention of coherent structures will lead to a reduction of aerodynamic drag. However,the dynamics of coherent structures may partially depend on the form of the bluff body. Thisstudy focuses on the geometry of a squareback vehicle and the detailed observation of occuringcoherent motions of fluctuations in the wake.Simliar to two-dimensional flows, mainly 3 different forms of coherent structures exist in thewake of a squareback vehicle based on the time and length scales. Firstly, vortices roll upemanating from the Kelvin-Helmholtz (KH) instability in the shear layer. Secondly, compar-atively larger vortices shed from the dead water region. The vortex shedding occurs in timescales of one order of magnitude above the formation of KH vortices. Thirdly, within timescales of at least one order of magnitude above vortex shedding, the entire dead water regionexperiences deflections. The following paragraphs summarize the documented observations ofthese coherent structures on vehicles and vehicle models so far.Kelvin Helmholtz (KH) vortices emerge in the shear layer of almost every wake flow. Barroset al. (2015) detects frequencies between Sr = f δ S /u = 0 .
23 and Sr = 0 .
29 (related tothe local shear layer thickness δ S and the velocity u = 0 . u ∞ , with the freestream velocity u ∞ ) of an Ahmed vehicle model in a Reynolds-numbers range of Re h = u ∞ h/ν = 2 10 and Re h = 4 10 (with the vehicle height h and the kinematic viscosity ν ). The peaks ofthe KH mode are broadband and of decreasing frequency with increasing distance from theseparation edge (Barros et al., 2015). The decrease of frequency with distance is interpreted asa consequence of the vortex pairing. Grandemange et al. (2014) investigates the shear layersin the wakes of two production cars at a Reynolds-number of Re l = 10 . These investigationsshow the phenomenon of vortex pairing of shear layer vortices in terms of a linear growth ofshear layer in flow direction and proportional decrease of frequency of the dominant part offluctuations.Vortices are permanently shed from the near wake into the far wake. Publications related tosquareback vehicle models of Grandemange et al. (2013b) at Reynolds-number Re h = 9 . and Barros et al. (2015) at Reynolds-number Re h = 3 10 detect the frequency of vortexshedding between Sr h = f h/u ∞ = 0 .
127 und Sr h = 0 . oherent Structures in a Vehicle Wake 3 ative to the mean flow direction. Up to now the literature has reported these low frequencyfluctuations of the wake only in terms of a symmetry breaking mode through a bi-stability(Grandemange et al., 2013b,a; Brackston et al., 2016). In case of a bi-stable behavior the wakeis deflected and switches direction sponaneously after long time scales (compared to the typicaltime scales of the vortex shedding (see Grandemange et al., 2013b)). The averaged timescaleof the switches is approximately ∆ tu ∞ /h = 1500 and scales with the velocity. The geometricproportions of the vehicle like the ratio of height to width h / b and the ground clearance de-termine whether a bi-stability occurs and the axis of the bi-stability is aligned with the heightor the width of the vehicle (Grandemange et al., 2013a). The investigations of Grandemangeet al. (2014) on 2 production cars and those of Cadot et al. (2016) on 4 production cars fora Reynolds-number of Re l = u ∞ l/ν = 10 (with the vehicle length l) show that bi- stabilitiesexist under realistic conditions. However, they also show that they may not always occur.Some approaches of active flow control reveal aspects of the behavior, the structures andlocal phase relations of detached vortex structures in the wake of squareback vehicles. Theresponse of symmetric and antisymmetric excitation on the separation edges of an Ahmed modelwith synthetic jets confirms the vortex shedding in a meandering form Barros et al. (2016).Rigas et al. (2014, 2017) observe a comparable response for axialsymmetric bluff bodies. Theexcitation of the vortex shedding results in a reduction of fluctuating energy for other coherentstructures (Rigas et al., 2017). This indicates that the dynamics of different coherent structuresare affecting each other.There are still many geometrical influences where only little is known about the effect oncoherent structures and the vortex shedding. Some of these are the front shape (e. g. withoutseparation bubble or rather sharp front shape), a rear diffuser or the moving ground. Barroset al. (2016) mention in their work that further studies for the influence of different geometries(e.g. aspect ratio of the base, ground clearance) are necessary for a better understanding ofthe vortex shedding. There is also no reported study for the fluctuation motions at long timescales in the case when no bi-stable behavior is detected. However, the occurence of a bi-stablebehavior depends on geometrical influences (Grandemange et al., 2013a). Hence the questionconcerning the influence of geometrical parameters on vortex shedding can be extended to lowfrequency behavior.With respect to flow control of fluctuating motions the understanding of processes in thewake flow and their relation to the behavior of base pressure fluctuations is important. Thisapplies especially to possible different types of behaviors dependent on geometrical parameters.An example of an effective control of coherent structures in two-dimensional flow shows theimportance of the understanding of this relation (Pastoor et al., 2008). Recent results on activeflow control on three-dimensional bluff bodies (Rigas et al., 2017) and on an Ahmed squarebackmodel (Barros et al., 2016) support this statement. These studies show that the understandingof coherent structures and their phase relation on different positions as well as the directionof motion are essential to create concepts for an effective control. Furthermore, the exampleof the two-dimensional flow shows particularly that the understanding of the relation betweendynamics of coherent structures and the resulting base pressure behavior can be valuable forsensor design and position for an effective feedback loop flow control (Pastoor et al., 2008).This study investigates coherent structures in the wake of a SAE squareback vehicle modelas an approach to address these open questions. The focus is on the vortex shedding and fluc-tuating motions at low frequencies. The coherent structures of such a model and its particulargeometric characteristics have not been documented until now. The SAE squareback modelrepresents a geometry with a rear diffuser and with an aspect ratio of the base that differs from B.Bock the geometries that were investigated so far (Grandemange et al., 2013b; Duell and George,1999). Together with these results the present study gives an impression of the possible impactof these geometrical details on coherent structures.The present work is structured as follows. Section 2 presents methods of filtering and separat-ing coherent structures as well as the motion of structures from other remaining non-coherentturbulent fluctuation motions. The following section describes the experimental setup and themeasurement configuration for data acquisition. Section 4 introduces the numerical model thatis used to extend the experimental observation of results. The results of the study are presentedin section 5. First the results on base pressure and then on the flow field are shown. Theseresults are further subdivided by different time and length scales. In the discussion in section 6these results are compared to findings from other investigations and extended to more typicalproduction car shapes provided by additional experimental data. The results of this presentstudy are summarized in section 7. This last section provides an outlook related to the impacton the initial questions in section 1 and concludes with ideas for further research to answerremaining questions.
The investigation of coherent structures in wakes requires a concept for coherent structuresas well as the application of methods that allow a distinction of different coherent structures.This section discusses the interpretation of coherent structures in the present investigation.The remaining part of this section is dedicated to the applied methods for an identification anda description of coherent structures.
Coherent structures are related to large scale vortex-like structures. However, the description ofcoherent structures differs in some details from the definition of vortices and turbulence. Manyexamples for the utilization of the concept of coherent structures can be found in research relatedto turbulent flows (Berger et al., 1990; Hussain, 1983, 1986; Lee et al., 2004; Rempfer and Fasel,1991; Pastoor et al., 2008; Pujals et al., 2010; Sirovich, 1987; Fuchs et al., 1979; Adrian et al.,2000). The example of the intuitive definition of a vortex by closed, circular pathlines arounda center only leads in some special cases to the identification of vortices (Hussain, 1986; Adrianet al., 2000). The weak definition of coherent structures offers many possibilities compared tothe definition of a vortex. This serves as a possibility to a simpler access in their observation.To clarify the description of coherent structures for the present study the following list gives aweak definition of the term. This definition follows mainly Hussain (1986) and Ho and Huerre(1984). Coherent structures are essentially related to: • The occurrence of a temporal or spatial coherence, orderliness or structured behaviorrelative to the temporal or spatial average. • Vorticity, or vortical structures. • Fractions of large scale or low frequency fluctuations in the turbulence spectra (incomparison to the dissipative scales). • A spatial coherence that may appear as an unsteady, periodic but also intermittent pro-cess. oherent Structures in a Vehicle Wake 5
The following part of the section presents methods for the description and observation ofcoherent structures.
Small scale fluctuations of a pressure field are separated from large scale fluctuations by ob-servation of the barycenter (equation 1, (see Grandemange et al., 2013b)) or a dimensionless,spatially averaged surface pressure gradient (equation 2, (see Grandemange et al., 2013a)). Thisresults in two values for each quantity (2 coordinates or directions of the gradient) for a givenpressure distribution. These values characterize the considered domain in terms of large scalemotions. The barycenter is determined by the quotient of the integral from the product of thelocal base pressure and its position, divided by the integral of the base pressure. The dimen-sionless, spatially averaged surface pressure gradient is computed by the sum of the product oflocal average of pressure along a constant coordinate c pz,i or c py,i with the local coordinate y i , z i relative to the maximal distance of measurement point locations ∆ y max , ∆ z max . y p = (cid:82) yc p ds (cid:82) c p ds ; z p = (cid:82) zc p ds (cid:82) c p ds ; (1) ∂c p ∂y = Σ c pz,i y i ∆ y max ; ∂c p ∂z = Σ c pz,i y i ∆ z max ; (2)Periodically occuring processes are considered by power spectral densities (PSD, S xx ( f )) infrequency space. Hence, period times T represent frequencies f. The PSD is computed usingthe autocorrelation function r xx ( τ ) (equation 3) of the time row x ( t ) which folds the time rowto different time lags τ . The PSD (equation 4) is calculated by Fourier transformation ofthe autocorrelation function r xx ( τ ) . The impact of random fluctuations is reduced throughaveraging over overlapping windows of autocorrelation functions. Spectral leakage through thelimited window is avoided by window functions. r xx ( τ ) = lim T →∞ (cid:90) T − T x ( t ) x ( t + τ ) dt (3) S xx ( f ) = 12 π (cid:90) r xx ( τ ) e − j πfτ dτ (4)To assess peak frequencies the frequency f must be available in a meaningful format. This isthe Strouhal number (equation 5) Sr h = f hu ∞ (5) The Proper Orthogonal Decomposition (POD) is utilized here to identify spatial structures andrelated time scales as well as for a Low Order Modelling (LOM). The separation of coherentstructures is based on the POD method of snapshots (Sirovich, 1987). Starting point is thedecomposition of fields from a number M of snapshots u ( x, t ) into fluctuating fields u (cid:48) ( x, t ) andan averaged field ¯ u ( x ). The POD decomposes the fluctuating fields in M spatial modes φ m ( x )and temporal coefficients a m ( t ): B.Bock u (cid:48) ( x, t ) = M (cid:88) k =1 a m ( t ) φ m ( x ) . (6)The spatial modes describe the spatial distribution of the fluctuations and the temporal coeffi-cients reflect their time-dependent amplitude. The modes φ m ( x ) correspond to the eigenvectorsfrom the Singular Value Decomposition (SVD) of the autocorrelation matrix R ( x ). R ( x ) = u (cid:48) ( x, t ) · u (cid:48) ( x (cid:48) , t ) . (7)The eigenvalues λ k of the SVD represent the energy fraction of the turbulent kinetic energyof the fluctuations. In the process of the SVD, the eigenvalues are normalized, so that (cid:107) φ k (cid:107) = 1and the modes are sorted by descending size (energy fraction). The temporal coefficients aredetermined by the projection of the fluctuations on the modes: a k ( t ) = φ k ( x ) × u (cid:48) ( x, t ) . (8)The time coefficients are considered in a spectral decomposition to analyze the time scales ofthe modes. In addition, the spatial, time-dependent behavior and fluctuations of the modes areinvestigated. Lower Order Models (LOM) are used here. A LOM can consist of one or moremodes with m = 1 to m = M . Equation 9 represents the reconstruction of a field with a LOM: u LOM ( x, c ) = ¯ u ( x ) + M (cid:88) m =1 c m φ m ( (cid:126)x ) . (9)The averaged field ¯ u ( x ) is superimposed with the product from the mode φ m and the am-plitude c m for all modes taken into account. The limitation of the number of modes can beconsidered as a spatial filter. Since spatially small structures in the turbulent spectra usuallycorrespond to a small energy fraction of the fluctuations, higher mode numbers m tend to rep-resent smaller spatial structures (and higher frequencies). In the simplest case, the amplitude c m corresponds to a selected amplitude a m ( t ) at a selected time instant t . In more complexLOMs, an appropriate definition must be given.Mode structures of two modes, which represent the same size of structures shifted by a quarterof a wavelength in the flow direction, are interpreted and modeled as a convective mode pair. Intime-resolved data sets, the same time scales occur for these modes in convective mode pairs inthe time coefficients. This is used for the LOM to describe a process and define the amplitudes c m . For the LOM of a convective mode pair, the phase definition based on the time amplitudes a ( t ) and a ( t ) as used by Chiekh et al. (2013) and given by equation 10 is applied: ϕ , ( t ) = arctan √ λ a ( t ) √ λ a ( t ) . (10)For eigenvalue based, phase-dependent amplitudes, the following applies: c ( t ) = (cid:112) λ sin ϕ , ( t ) , c ( t ) = (cid:112) λ sin ϕ , ( t ) . (11)Due to disordered, turbulent and intermittent fluctuating motions, different amplitudes c ( t ), c ( t ) exist for the same phase ϕ ( t ) at different times t . A phase-averaged amplitude c , c canbe calculated, which can be used as LOM to describe a phase-averaged process. In some cases, oherent Structures in a Vehicle Wake 7 the higher order modes m = 3 and m = 4 are also included in the phase averaging based onthe phase definition of equation 10: c ( t ) = (cid:112) λ sin ϕ , ( t ) , c ( t ) = (cid:112) λ sin ϕ , ( t ) . (12) The coherent structures in the wake of an SAE model were experimentally investigated in awind tunnel. The geometry of the SAE squareback model is described by Lindener (1999). Thewind tunnel design, measurement technology and data recording used are described below.
Figure 1: Experimental setup of the SAE squareback model in the wind tunnelThe experimental investigations on a 1 : 4 SAE squareback model were carried out in themodel wind tunnel of the Institute for Internal Combustion Engines and Automotive Engineer-ing (in German Institut f¨ur Verbrennungsmotoren und Kraftfahrwesen: IVK) at the Universityof Stuttgart. The wind tunnel is equipped with a system for simulating the moving ground,which includes a five belt system and a boundary layer pre-suction system. A block profile ofthe flow is generated with the boundary layer pre-suction and the moving floor. For the velocityranges between u ∈ [30; 80] m s − the turbulence intensity is T u < .
3. Further details on thewind tunnel are described in Wiedemann and Potthoff (2003). The measurements were carriedout for the velocities u = { , , , , } m s − and correspond to the Reynolds number Re h = { . , . , . , . , . } . In this publication only the results at Re h = 1 . are de-scribed. However, the same behavior was observed for all Reynolds numbers. The arrangementof the model and the measuring technique is shown schematically in figure 1. The picture showsthe nozzle outlet on the left and the collector on the right with the SAE model in between. B.Bock
The coordinate system is defined in such a way that the x -axis runs parallel to the flow, theand the z -axis runs parallel to the height direction with the origin in the middle of the basesurface.The reference velocity u ∞ indicates the calibrated wind tunnel velocity without the model andcorrected by wind tunnel interference effects. The interference effects were corrected accordingto the methods proposed by Mercker et al. (1997) and Mercker and Wiedemann (1996). Thecorrection factors c q = 1 .
01 for the dynamic pressure and c u = 1 .
005 for the velocity weredetermined through this method. The correction factors are used in the sense of global valuesfor the reference velocity u ∞ and for quantitative comparisons of the base pressure. A localcorrection would be necessary for field data. However, this is only necessary for quantitativecomparison of the absolute values and is therefore not used here. Time-resolved measurement data of the effects of the coherent structures on the base wererecorded using pressure probes. The black dots in Figure 1 at the base mark the pres-sure measurement points. The base pressures were measured at 24 points using the pres-sure transducers marked in Figure 1. The pressure transducers were placed at coordinates y/h = {− . , − . , − . , . , . , . } , and z/h = {− . , − . , . , . } . Eachpressure transducer has an internal diameter of d a = 0 . . m long pipeline with an internal diameterof 1 . mm . The pressure transducers are calibrated and have a temperature compensation.The operating range of the pressure transducer is up to 6 . kP a with an accuracy of 0.05%. The sampling rate is f s = 250 Hz (sampling dimensionless frequency or Strouhal number S s = { . , . , . , . , . } ). The resulting maximum frequency is expected to be at leastcapable of detecting vortex shedding signals (if present at this location) which are expected to bein dimensionless frequencies between ∆ Sr h = 0 . Sr h = 0 .
25 (compare section 1). In or-der to take dynamic effects of the system of pipeline and measuring volume of the pressure trans-ducer into account, a dynamic correction of the measured pressure values according to Berghand Tijdeman (1965) was applied. Each pressure transducer was recorded over a total durationof 90 seconds. This corresponds to about 22103 measured values or time instants per pressuretransducer and a dimensionless duration of tu ∞ /h = { , , , , } con-vective units. These timescales are expected to cover the switches of bi-stable behaviour (seesection 1). Pressure readings are represented as dimensionless values c p by the pressure differ-ence to the plenum pressure p ∞ of the wind tunnel related to the dynamic pressure q ∞ = 0 . u ∞ .Time series spectra were used as Power Spectral Densities (PSD) with a Chebyshev windowof size 150 and 50 % overlap. This results in a frequency resolution of ∆ f = 1 . Hz , or∆ Sr h = { . , . , . , . , . } .In order to obtain time-resolved data of the flow field, additional point measurements werecarried out in the shear layer at a Reynolds number of Re h = 1 . . For this purpose, amulti-hole cobra probe of the manufacturer TFI with a scanning frequency of f s = 3500 Hz ( Sr s = 21) was used to measure for 90 seconds ( tu ∞ /h = 15 10 convective units). Thus thetime-resolved velocity components were measured at 24 measuring points in the shear layer atthe positions x/h = { . , . , . , . , . , . , . , . , . , . , . , . } with the cor-responding coordinates z/h = { . , . , . , . , . , . , . , . , . , . , . , . } in the upper shear layer behind the separation edge. The PSD was used for these signalswith a Hanning window of size 512 and 50 % overlap. This leads to a frequency resolution of oherent Structures in a Vehicle Wake 9 f = 3 . Hz , or Sr h = 0 .
02. The resulting maximum frequency for the shear layer measure-ments are expected to be at capable of resolving KH vortex signals (if present) after paring,e.g. in regions where the shear layer thickness is thick enough. For example for a shear layerthickness of δ S = 0 . h a dimensionless frequency of ∆ Sr h = 1 .
933 (compare section 1) or less isexpected. This should be sufficient to follow at least to some extend the decrease of frequencythrough the vortex pairing along the growing shear layer.
Particle Image Velocimetry (PIV) was used to observe the flow around the near wake. Forthis purpose a Nd:YAG laser with a fixed frequency and thus a sampling rate of 10 . Hz , awavelength of 532 nm , and a maximum energy of 850 mJ per pulse was used. The green areasor borders in Figure 1 represent the laser light sheet and the parallel staggered green areas thelight sheets in the y -plane around the wake of the model. The thickness of the light sheet wasabout 2 mm . The incoming flow was mixed with aerosol droplets of Di-Ethyl-Hexyl-Sebacat(DEHS) with a diameter of about 1 µm . With two cameras of the type Imager sCMOS with aresolution of 2560 × − ◦ / 11 ◦ to the x -axis in the z -plane, with a distance of2 m to the light sheet plane and equipped with a 50 mm lens. The resulting field of view was175 × mm . The time delay of the double images was 22 µs . For the temporal averaging 300snapshots were recorded. The velocity vectors were calculated with an interrogation windowof 32 ×
32 with 75 overlaps. The resolution is 1 . mm or 0 . h . No smoothing or filter wasused in the evaluation. Several parallel y -planes of the wake were measured (see Bock, 2019).In this publication only the y = 0 plane is considered.The sample rate for PIV is lower than the frequencies of interest for the vortex shedding andthe KH vortices. For low frequency dynamics the measurement time is too short. ThereforePIV results are considered here for the structures of dominant fluctuating motions, based ona statistical (since not time resolved) representation of the wake dynamics. To observe thefrequencies of the wake dynamics pressure measurements in the wake and at the base are con-sidered. Complementary, simulation results provide a basis for combined resolved frequenciesand dynamic structures for the modes of interest. The measurement data were extended by data from a simulation using the commercial soft-ware package SIMULIA PowerFLOW R (cid:13) version 4.3d, which is based on the Lattice-Boltzmannmethod. Turbulence is treated using the Very Large Eddy Simulation (VLES) approach. Theunresolved turbulent scales are represented by a variant of the RNG k model (Yakhot andOrszag, 1986). The tangential velocity component on all friction walls is approximated withan extended standard model that includes the influence of the pressure gradient and surfaceroughness (Launder and Spalding, 1974). The domain of flow simulation consists of a box around the SAE model. The domain coversa volume of 47 h × h × h (14 . m × . m × . m ). The coordinate system is defined according to the experimental setup so that the x -axis runs parallel to the flow and the z -axisruns parallel to the height direction with the origin in the middle of the base surface. Theinlet boundary condition with the defined flow velocity of u ∞ = 50 ms − (corresponds to Re h = 1 . ) is 20 . h (6 . m ) ahead of the leading edge of the model. As outflow boundarycondition a constant pressure of p ∞ = 101300 P a is imposed. A boundary condition of afriction-less wall (or also symmetry boundary condition) was applied to the lateral and theupper boundary of the domain. The vehicle itself is represented by a friction wall boundarycondition in the simulation. The discretization is realized by a mesh of cubic simulation voxels.In 10 resolution domains ( rd ), the resolution is halved from one level to the next in relationto the vehicle. The coarsest step has a resolution of rd/h = 2 . u ∞ . In offsets of the vehicle surface of { } the resolution levels of rd/h = { . , . , . , . , . } were defined. Theseparation edge and the wake of the models were resolved with rd/h = 0 .
004 in a domainof 2 . h × . h × . h starting with x = − .
066 h from the base, with the ground andsymmetrically around y . Around the A-pillars of the model, a further offset area by 0 . h was resolved with rd/h = 0 . t = 1 . s . Thesimulation was simulated for tu ∞ /h = 1043 convective units (corresponds to 6 .
26 seconds)and was initialized with the solution of a previous simulation. Detection of the end of thesimulation start-up process was performed according to the method proposed by Mockett et al.(2010) based on the time series of the drag coefficient. The settling time is tu ∞ /h = 16 . . tu ∞ /h > .
16 seconds). In Bock (2019) time-averageddata from measurement and simulation can be compared. The qualitative agreement of thedistribution of velocities and velocity fluctuations in the wake, as well as the base pressures andthe base pressure fluctuations is very good.
Synchronized recordings of the base pressure and the u , v , w velocity components (correspond-ing to x -, y -, z -direction) as well as the total pressure p t of the flow field are used to observe thecoherent structures. The wake volume was recorded with a spatial resolution of rd/h = 0 . f s = 100 Hz ( Sr h = 0 .
6) with a moving average over this period for a duration of tu ∞ /h = 333convective units (2 seconds) and thus comprises 200 snapshots of the volume. This record-ing is capable to resolve time scales of dimensionless frequencies between ∆ Sr h = 0 . Sr h = 0 .
25 expected for the vortex shedding (compare section 1).The low-frequency scales were recorded at a rate of f s = 30 Hz ( Sr h = 0 .
18) with a movingaverage over this period for a duration of tu ∞ /h = 1000 convective units (6 seconds). This dataset thus contains 180 snapshots of the volume. Moving average is a filter of higher frequencytime scales and thus of smaller length scales. This enables a clear representation of large-scalecoherent structures. For the calculation of the PSD, a Chebyshev window of length 40 with50 % overlap, resulting in a frequency resolution of f = 2 . Hz ( Sr h = 0 . f = 1 Hz ( Sr h = 0 .
06) was oherent Structures in a Vehicle Wake 11 used.
In the presented results, the coherent structures in the postprocessing of an SAE vehicle modelfrom the measurement and simulation data are described and documented using the methodsmentioned above. The results refer to the Reynolds number Re h = 1 . , whereby the samerepresentations of the experimental data also apply to all other Reynolds numbers investigated.The results will be presented in the following order. At the beginning, the averaged flow fieldis observed. Some special features of the wake topology, i.e. the vortex ring, are highlightedin comparison to other documented vehicle models. The subsequent considerations of the basepressure distribution and the motion of the barycenter of pressure give first insights into theareas with the strongest energy losses (in the form of low base pressures) and the directions ofmotion of the wake. The frequencies of the dimensionless, spatially averaged surface pressuregradient (equation 2) and the velocity fluctuations in the shear layer give an impression of thetime scales of the motions in the wake. Thereupon the motions in the middle frequency rangeof the wake in the flow field are considered. In this frequency range a vortex separation processis to be expected, which is then described in more detail with this observation. Finally, the lowfrequency motions of the flow field in the wake are considered. Figure 2: Averaged base pressure (left top), base pressure fluctuations (left below) from exper-iments and visualization of the wake flow around the SAE squareback model withisosurface of c p,t = − .
23 from simulationThe dead water of the SAE fullback model consists of a vortex ring which is deformed in theupper section of the dead water by the influence of the diffuser and moves closer to the base inthe lower section. The shape of the vortex ring and thus the dead water area differs from theshape of the vortex ring on an Ahmed squareback model. Figure 2 describes on the right handside the flow volume in the wake based on the simulation data. The right figure shows the rearend of the model (in grey) and the streamlines in the middle plane of the wake with coloring of the total pressure coefficient, as well as the isosurface of the total pressure c pt = − . Snapshots of the pressure distribution show mainly deflections of the distribution in the widthand the lowest pressures in the lower range in z -direction. The motions of the wake can beanalyzed with the distribution of the base pressure. The higher sampling rate of the pressuremeasurements compared to the flow field measurements can be used as an advantage in termsof temporal resolution of coherent structures. Figure 3 illustrates some basic properties of thisobservation in the form of two snapshots. The images of the two snapshots show the y -direction(width) above the abscissa and the z -direction (height) above the ordinate. The separation edgeis shown as a black thick line. Grey crosses mark the pressure measuring points. The pressuredistribution is shown by the coloring. A barycenter of the lowest pressure is calculated for therespective distribution of the time instant and displayed as a black × to demonstrate how thedetermined pressure center corresponds to the pressure distribution. The left snapshot showsan asymmetric distribution. In principle, the pressures in the lower ranges ( z/h < − .
4) are lowfor both snapshots. At the asymmetric distribution (left side) pressures at right upper quarterare high. The symmetrical distribution shows the highest pressures in the upper, middle area,which decreases slightly outwards and slightly more downwards. The calculated focal pointscorrespond well with the visual distribution and are therefore suitable for further evaluations.The barycenter of the low pressures lies in the lower half and moves mainly in the width.Figure 4 shows the distribution of the barycenter positions from snapshots of the pressuredistribution. The rear edge is marked with a thick black line, the y -direction is plotted on theabscissa and the z -direction on the ordinate. It is shown that the barycenter and thus alsothe wake moves primarily in the y -direction and comparatively little in the z -direction (i.e.height). It should be noted that the barycenter is in the lower half. However, the motions of oherent Structures in a Vehicle Wake 13 the barycenter are due to the fluctuations in the upper half, as the fluctuations in this areadominate (see Figure 2).Figure 3: Snapshots of base pressure distributions from experimentsThe distribution of the snapshots of the barycenters of the base pressure distribution indicatea lateral deflection of the wake and thereby exclude a bi-stability. The distribution of thepressure points of the individual snapshots confirms these findings. The pressure centers arerepresented by rings with the indicated rear contour in Figure 4 above. Most of the barycentersfrom the series of 5000 snapshots are located in the lower half of the base. The points ofthese snapshots correspond to tu ∞ /h = 3300 convective units which should be long enough toexperience a bi-stable switch of lateral deflection. Even considering the complete series of 22103snapshots ( tu ∞ /h = 15000) the distribution of pressure centers do not change. Instead, thefocal points are distributed over a large span along the width. From the shift of the barycenterit can be concluded that the wake is deflected more in y - than in z -direction. Figure 4 belowshows the frequency distribution of these positions along the width. The distribution of thebarycenters shows an obliquity. However, the focus is clearly on one position and not two. Abi-stable behavior can thus be excluded as the cause of the lateral deflections of the wake.Figure 4: Distribution of barycenters of pressures from snapshots on the base surface (top) anddistribution function along y from experiments. The base pressure fluctuations are mainly dominated by the effects of the deflection of thewake in the direction of the width with a low frequency. However, the effects of deflectionsin width and height direction in the medium frequency range can also be determined in thebase pressures. Figure 5 shows the PSD of the time series of the dimensionless, spatially av-eraged surface pressure gradient in y - (top) and z - (bottom) direction for different Reynoldsnumbers. In the spectra, very low frequencies ( Sr h < .
03) of the motions in y- and z- direc-tion, but especially in y- direction, are dominant. For the y- position, a survey or at least aplateau at the frequency Sr h ≈ . ... .
18 is also recognizable for all Reynolds numbers. Inz-direction, the barycenter of pressure is deflected less severe. Nevertheless, the fluctuations at Sr h ≈ . ... .
19 are also discernible in z -direction. All raised areas appear as very broadbanddistributed fluctuations. The strongest deflection occurs at low frequencies. However, there isalso a deflection of the wake, which moves in both directions with time scales that correspondapproximately to dimensionless frequencies of Sr h ≈ . ... .
19. The spectra of some individualpressure measurement points (not shown here) also show these frequencies (see Bock, 2019).Figure 5: Spectra of the dimensionless, spatially averaged surface pressure gradient on the basesurface in y- (top) and z- direction (bottom) at different Reynolds numbers fromexperiments.
In the vicinity of the separation edge, a decreasing frequency peak at high frequencies correlateswith the increase in the shear layer thickness. Spectra from the time series of measuring pointsfrom the upper free streamline of the shear layer in the center plane ( y/h = 0) are shown inFigure 6. The PSD of the u (cid:48) y fluctuating velocity component is plotted over different positionson the abscissa, while the dimensionless frequency Sr h is plotted on the ordinate. The PSD is oherent Structures in a Vehicle Wake 15 indicated by the coloring with logarithmic scaling. The fluctuations in the upper shear layerare highest (cf. Figure 2). For this reason, the lower shear layer is not shown here. However,they can be seen in the work of Bock (2019). In principle, distributions of the fluctuationsof the upper and lower shear layer are similar. A feature of these spectra is the shift of abroadband peak near the base from high frequencies to low frequencies further downstream.The first occurence is recognizable at x/h = 0 .
35 with Sr h = 2, which changes up to Sr h ≈ . x/h = 0 .
8. This transition is marked in the illustration with a dashed, diagonal line. Thisprocess describes a drop in the characteristic frequency f c ∼ u /δ S in the direction of flow,which is also characteristic of free shear layers. In this relation δ S is the shear layer thickness.The velocity u represents the time averaged flow velocity in the center line of the shear layer.This velocity undergoes only minor changes in the considered range of x/h (see Bock, 2020).The almost linear decrease of the frequency f c is caused by the rolling up and thus the growthof KH vortices in downstream flow direction in the shear layer. In the work of Bock (2020) alinear increase of the shear layer with δ S /dx ≈ . δ S /dx = 0 . ... . δ S /dx = 0 .
14, or δ m /dx = 0 .
12 forthe vehicle types Renault Trafic, respectively Peugeot 3008. The behavior of this high-frequencypeak is also consistent with the observations of Duell and George (1999) on a squareback model.Figure 6: Spectra (fluctuating part of PSD: u (cid:48) y ) over x-position along the upper free streamlinein the wake plane y/h = 0 from experiments.The strongest fluctuations in the shear layer mainly occur with Sr h ≈ . x/h ≈ . ... . x/h > .
8. This correlates with the observations of the fluctuationsin the flow field in Figure 2. The broadband distributed fluctuations increase, with maximumdistance to the base for example at x/h = 1 . Sr h ≈ .
22, which is drawnas a horizontal dashed line. The accumulation of fluctuations around Sr h ≈ .
22 can still beseen further downstream.
In this work intermediate frequencies are considered as frequencies lower than frequencies ofKH vortices, related to vortex shedding with vortices larger than the KH vortices. Snapshots of the dead water show vortex structures of different sizes, especially in the area of the upper shearlayer close to the base. A first impression of the coherent structures results from a few snapshotsof the flow field. As an example, Figure 7 shows two snapshots experimentally determined forthe center plane ( y/h = 0). The illustration shows the rear edge of the model in grey and thestreamlines in the plane of the wake. In the lower domain ( z/h < z/h >
0) curved streamlines arealso recognizable. These are mainly located at the boundary with rather straight streamlines,i.e. in the shear layer. In the left part of Figure 7, there are curved (but not closed) streamlinesin the upper section at x/h ≈ .
25 and x/h ≈ .
5. At x/h ≈ . x/h ≈ . x/h ≈ .
3. Both vortices rotate in clockwise direction and correspondto the expected direction of rotation in the upper shear layer. The streamlines starting fromthe lower and upper separation edge meet at the instantaneous position of the saddle point atabout x/h ≈ . y/h = 0 from experiments.A more detailed description of the processes and the spatial scales is enabled by the recon-struction from the phase references of the first mode pair of the POD. The first two POD modesof the flow field represent a convective mode pair and indicate a vortex separation process. Fig-ure 8 shows the distribution of the vorticity in the wake of the SAE model in the y/h = 0 planeof the POD modes m = 1 (left) and m = 2 (right). The two POD modes of the flow field repre-sent structures of the same size and wavelength, which are only spatially shifted by a quarter ofa wavelength. These structures are also shown at other measured planes (not shown here, (seeBock, 2019)). Thus a convective mode pair can be assumed. A phase-averaged reconstructionbased on the phase definition between this mode pair can be conducted (see equation 10).In reconstructions of POD of individual planes, a vortex motion occurs as a global mode inthe upper dead water. The streamlines from the reconstructed flow fields can be seen in Figure9. This reconstruction of the wake plane represents the flow in the center ( y/h = 0) based on thephase definition with modes m = 1 and m = 2 from the POD. The phase position was shiftedby ∆ ϕ = 113 ◦ without affecting the generality of the results. The illustration shows streamlinesand the rear of the model in grey. Starting with the phase ϕ = 0 ◦ (above, left) vortices arevisible in the dead water, which in principle resemble the arrangement of the averaged flow(Figure 2 right side), with a larger lower vortex close to the base and a smaller upper vortex oherent Structures in a Vehicle Wake 17 which is more downstream to the base. It is noticeable in this phase that the upper vortex islarger compared to the averaged flow. Connected to this observation, the position of the saddlepoint is slightly shifted downwards. The saddle point (in the figure marked by a red dot) isinterpreted as an indicator for the length of the dead water region. In the following phase ϕ = 90 ◦ the upper vortex becomes larger in height (z - direction) and its center moves towardsthe base. This shifts the saddle point upwards. At the same time, the streamlines behind thesaddle point are deflected upwards. As in the following phases, the lower vortex hardly changes,at ϕ = 180 ◦ the upper vortex becomes very small in the dead water. However, curved, butnot closed streamlines can be seen behind the saddle point at x/h ≈ .
2. This indicates vortexstructures downstream of the dead water. In phase ϕ = 270 ◦ the upper vortex in the deadwater can no longer be identified. The saddle point as convergence point of the streamlinesof the upper and lower wake area must be either significantly closer to or significantly furtheraway from the base surface than in the other phases. It should be noted that the lower freestreamline (which separates the outer flow from the dead water) starts above z/h > − .
5, asthe rear diffuser is located in this plane. At x/h ≈ . m = 1 (left) and m = 2 (right) in the wake plane y/h = 0 from experiments.In the work of Bock (2019) these processes are shown for further wake planes by modestructures and by streamlines of their reconstruction from experiments. It can be seen thatthe modes m = 1, m = 2 in the other planes are convective modes and that they behavevery similarly, taking into account the fundamental change in the temporally averaged flowover the width. Nevertheless, with the evaluation of individual planes, ordered processes acrossthe planes can hardly be considered. To illustrate the three-dimensional flow of the process inFigure 9, POD mode decomposition from the simulation is used below.The POD modes from m = 1 to m = 4 of the high frequency flow volume recording fromthe simulation represent two convective mode pairs that are temporally and spatially paired.The behavior of the dominant fluctuation motions in the context of the considered mean timescales can thus be modelled from these 4 modes. The composition of POD modes in the wakevolume differs in some small details from the POD modes of the plane flow field from themeasurement. This can be due to the different sampling rates, but also to the observation ofthe correlations in the flow field over the entire volume instead of just one plane. However,the structure of POD modes m = 1 and m = 2 of the flow volume from simulation is similarto Figure 8. Hence they represent a convective mode pair. In the same way, mode m = 3and m = 4 describe a convective mode pair. Furthermore, the modes of the flow field in the y -plane demonstrate that the simulation results show the same mode structure with the samemagnitude scale as the experimentally determined modes. The amplitudes spectra of PODmodes from m = 1 to m = 4 shown in Figure 10 show very dominant peaks in the samefrequency range. Mode from m = 1 to m = 4 have the most dominant fluctuating parts in thefrequency band around the broadband peak of Sr h ≈ .
22. Modes m = 1, m = 2 also havestrong fluctuations at low frequencies. This constellation of convective mode pairs with thesame dominant frequencies justifies a reconstruction with the first 4 POD modes by a modelingbased on the phase definition between the first mode pair (see equation 10). In terms of themotions of the upper wake vortex part, the wake saddle point and the holding position of thelower vortex in the y/h = 0 plane (Figure 11 to Figure 12), this modeling shows the samebehavior as in the experiment (Figure 9) and therefore reinforces this approach.Figure 9: Reconstruction from two modes superimposed with the averaged flow field in the y/h = 0 plane (phase offset of ∆ ϕ = 113 ◦ based on the phase definition betweenamplitudes of the modes m = 1, m = 2) from experiments.Figure 10: Spectra (Fluctuating part of PSD: a m ) of amplitudes from POD in the wake volumefrom simulation.In the near wake volume, the global mode is represented by the vortex shedding with a oherent Structures in a Vehicle Wake 19 superimposed, global deflection of the wake in direction of the vehicle width. Figures 11 and 12show the reconstruction of the process with Sr h ≈ .
22 in the wake volume using four phases.In the pictures, divided into 2 rows, 2 phases with 3 different views are shown. These viewsof the y - (top, left), x - (top, right) and z - (bottom left) planes are intended to illustrate thethree-dimensional motion of the wake volume. In the views, the rear of the model is shownas a light gray geometry. An isosurface of total pressure c p,t = − . y -plane is marked with a red point. The saddle point isinterpreted as an indicator of the length of the dead water region. To support the interpretationof the motion, the streamlines are shown in a plane in each view. The positions of these planes( y/h = 0, x/h = 1 . z/h = 0 .
35) are marked as dashed lines in the orthogonal views. Thephase states can be described as follows: ϕ = 0 ◦ : In the first row of Figure 11, the view of the y-plane is shown at the top and left. Inthe upper section of the dead water a vortex is located at x/h ≈ .
5. The saddle point in theplane of the streamlines is located in the upper half of the wake at x/h ≈ . z/h ≈ .
2. Inthe x-plane an asymmetric situation emerges. The streamlines of the x-plane show two vorticeswith their focuses located at z/h ≈ . / y/h ≈ + / − .
2. Accordingly, the total pressure inthis plane is distributed asymmetrically. On inspection of the streamlines in the dead water inthe z-plane this asymmetry is also visible. Streamlines of the z-plane (below, left) show twovortices. A large one at x/h ≈ . y/h ≈ − . x/h ≈ . y/h ≈ . ϕ = 90 ◦ : The upper part of the vortex ring has increased in height ( z -direction) in they-plane. The entire wake is more symmetric in all views and the streamlines behind the deadwater are hardly deflected in comparison to the previous and other phases. Thus the saddlepoint moves downwards ( z/h ≈
0) and closer to the base ( x/h ≈ . z/h ≈ y/h ≈ + / − . ϕ = 180 ◦ : The streamlines show that the upper part of the vortex ring in the y-planebecomes very small and shifts closer to the base again compared to other phases. Based onthe streamlines in this plane the saddle point is located at x/h ≈ . z/h > . x -plane an asymmetric situation appears again. As in ϕ = 0 ◦ the total pressure in this plane isdistributed asymmetrically. On inspection of the streamlines in the dead water in the z -planethis asymmetry is also visible. The vortices in the x-plane have changed to an asymmetricarrangement. Their focuses are located at z/h ≈ . y/h ≈ − .
45 and z/h ≈ . y/h ≈ . ϕ = 0 ◦ , the upper vortex is now on the right side. ϕ = 0 ◦ ϕ = 90 ◦ Figure 11: Reconstruction from the first 4 POD modes in superposition with time averagedflow. Isosurfaces of c p,t = − . oherent Structures in a Vehicle Wake 21 ϕ = 180 ◦ ϕ = 270 ◦ Figure 12: Reconstruction from the first 4 POD modes in superposition with time averagedflow. Isosurfaces of c p,t = − . ϕ = 270 ◦ : In the y-plane the saddle point has moved downwards and closer to the base( x/h ≈ . z/h ≈ . x/h ≈ .
25. A closer inspection of the upper part of the dead water of thisplane reveals another small vortex at x/h ≈ .
5. In comparison to phase ϕ = 180 ◦ the sizeof this vortex is increased. The location of the saddle point indicates that the downstreamvortex part has separated from the upstream part and left the boundary of the dead water. Aclue to a vortex shedding from the dead water. The streamlines and the isosurface of the totalpressure in the x-plane are more symmetric than phases ϕ = 0 ◦ and ϕ = 180 ◦ . However, at the x -plane (upside, right side) streamlines show two vortices asymmetric in size and position. Theisosurface of total pressure in the x -plane shows that the downstream section of the upper vortexis concentrated in the middle (in y -direction) area ( − . < y/h < .
25, 0 . < z/h < . − . < y/h < .
25. Similar observations can be seen bylooking at the different y -planes of the measurement (not shown here, (see Bock, 2019)).In the y -plane the limiting surface of the total pressure correlates well with the limitation ofthe dead water formed by the streamlines of the upper and lower sides in the y/h = 0 plane.Together with the upper vortex the dead water is squeezed relative to the base. However,elongated extensions of the isosurface of the total pressure are formed. These start from x/h ≈ z/h ≈ . x/h ≈ . z/h ≈ .
4. These extensions can also be observed inthe x -plane. There they are located at external positions y/h ≈ + / − .
4. The streamlines in thisplane show that they are surrounded by two vortices. In the x -plane the flow is symmetrical.In the z -plane the flow field (both isosurface and streamlines) is symmetrical as well. Theextensions of the isosurface of the total pressure are also enclosed by vortices in the z -plane.They thus form a horseshoe vortex which starts from the vortex ring in the dead water andcloses behind the dead water.Especially the streamlines in the y -plane allow the comparison with the phase states inFigure 9 and therefore a comparison of the reconstruction of simulation and experiment. Asthe phase states ϕ of experiment and simulation may not be perfectly synchronized, an exactmatch of the streamlines can not be expected. The phases considered in the simulation may liesomewhere between the phases shown in the experiment. Hence, they may represent transitionstates between the phases in experiments. However, the streamlines of the y -plane from theexperiments show that the main variations are caused by the changes of the size of the upperpart of the vortex ring in the dead water. The experimental results also show a related motionof the saddle point. In accordance with the simulation the saddle point moves up-/downwardsand away/towards the base during the process. In all observations this is accompanied by aformation of strong curvature of streamlines behind the saddle point. These curvatures indicatea vortex detached from the dead water. In conclusion a qualitatively good correspondence ofthe streamlines in the y -plane between experiment and simulation (cf. Figure 9 below left) isgiven. oherent Structures in a Vehicle Wake 23 maximal amplitudeminimal amplitude Figure 13: LOM by reconstruction of POD mode m = 1 of the wake volume sampled with f s = 30 Hz and in superposition with the averaged flow. Isosurfaces of c p,t = − . m = 1 from the POD decomposition of the long-termsampling at f s = 30 Hz ( Sr h = 0 .
18) and moving average from the simulation is used. Theinfluence of fluctuations from the high and medium frequency ranges is eliminated by the movingaverage. The temporal amplitudes of this mode fluctuate broadband in the range of the lowestresolved frequencies Sr h < .
04 as shown in Bock (2019). Figure 13 shows the reconstructionof the first mode in the wake volume using the maximum and minimum temporal amplitudesor a positive or negative deflection around the time-averaged flow field. In the picture, dividedinto two rows, the positive and negative deflections are shown with 2 different views each. Thereconstruction at maximum deflection of Mode m = 1 looks very similar to the averaged flow(cf. Figure 2) when viewed from the side. The upper part of the vortex ring is significantlysmaller than the lower part and at the same time has a larger distance to the base. Thusthe saddle point is at z/h ≈ . x -plane)a strong asymmetric flow field is shown. The two axial vortices in the considered plane areshifted to the left ( y/h < y -plane) of the reconstruction of theminimum deflection of mode m = 1, the flow field changes only insignificantly. In this case, the view of the x -plane corresponds to a mirroring around the y -plane. The data from the investigations in the wake of the SAE model show coherent structures ofdifferent time scales. The traces of the KH vortices in the velocity fluctuations of the shearlayer can be detected at a relatively high frequency. In addition, the vortex shedding in themedium frequency range can be traced back not only in the wake volume but also in the velocityfluctuations in the shear layer down to the base pressures. Moreover, a pronounced deflection ofthe wake can be recorded in all measurement data, which takes place in the very low frequencyrange. The interrelationships of the fluctuation motion of the vortex shedding and the deflectionof the wake in the width direction are discussed in more detail in the following.
The vortex shedding on the SAE model can be interpreted as a meander-shaped process inthe upper half of the wake. The changes in the flow fields in Figure 11 and Figure 12, whichtake place in time scales of Sr h ≈ .
22, can be interpreted as continuous process, where phase ϕ = 0 ◦ is followed by phases ϕ = 90 ◦ , ϕ = 180 ◦ , ϕ = 270 ◦ , ϕ = 0 ◦ , and so on.Starting from ϕ = 270 ◦ to ϕ = 0 ◦ , the size of the upper part of the vortex ring becomessmaller in the streamlines of the y/h = 0 plane. The upper part of the vortex ring becomeslarger in phase ϕ = 90 ◦ both in the streamlines of the y/h = 0 plane and in the isosurface of thetotal pressure. In the following phase ϕ = 180 ◦ it is again of reduced size. Hence, the vortexdetachment starts between ϕ = 180 ◦ and ϕ = 270 ◦ . The vortex shedding is connected with amotion of the barycenter of the total pressure in the upper domain from right ( y/h >
0) to left( y/h < x -planes. In phases ϕ = 90 ◦ and ϕ = 270 ◦ the flow field in this plane is symmetric. In theother two phases the total pressure and the streamlines are asymmetrically distributed. Fromvisual observations of the total pressure isosurface of phase ϕ = 0 ◦ a concentration of the upperpart of this surface in the right section ( y/h >
0) is concluded. In phase ϕ = 180 ◦ the totalpressure isosurface is concentrated in the opposing section ( y/h < y ≈ y -plane mark the shedding of a vortex during the same process.In phase ϕ = 270 ◦ branches of the isosurface of the total pressure at medium height ( z/h = 0 . x - and z -planes) indicate a horseshoe-shaped vortex.At a further state of the process ( ϕ = 270 ◦ ) this area within the total pressure isosurface isshown as a continuous area and branches move upwards ( z/h ≈ .
25) and become narrower (in y -direction). The switch of the concentration of the total pressure isosurface during the vortexshedding suggests a meandering detachment of the vortex of the upper part of the vortex ring.This describes a vortex shedding process in the wake. It is obvious that this is a vortex whichmaintains a connection through a horseshoe vortex during the shedding process from the vortexring (at least for a certain period of time). Such a behavior makes sense as Helmholtz’s lawof circulation remains fulfilled. With increasing distance from the base after leaving the deadwater, the detached vortex structure is more difficult to detect. This suggests the assumptionthat this will quickly vanish due to diffusion in the far wake. As a result, the vortex would haveto expand and lose intensity after leaving the dead water. oherent Structures in a Vehicle Wake 25 Figure 14: Schematic illustration of the vortex shedding at the SAE squareback vehicle.Vortex shedding on the SAE model can be considered as the motion of a vortex loop in theupper half of the wake. Figure 14 schematically shows the proposed vortex shedding in the wakeof the SAE model. The process is outlined in two views (top: y -plane and bottom: z -plane).The border of the dead water is shown as a black, solid line and the vortex ring is shown as blackcircles. Dashed lines represent borders of the vortices that do not lie in the plane of observation.At the y -plane a vortex starts to separate from the upper section of the dead water. Previouslydetached vortices can be seen further downstream. These vortices are getting bigger, weakerand increase their distance to each other the further they move downstream from the base.The increase in the distance of the vortices current distance is concluded from the differentfrequencies of the vortices (cf. Barros et al., 2015, 2017) of the vortex shedding depending ontheir orientation (interaction of the shear layers with the distance h or b ). The increasinggreying out of vortices embodies the decrease in the strength of the vortices. All detachedstructures are still connected to each other. By view from above, this connection representstwo vortices shifted in flow direction at opposite sides of the vehicle. These vortices withinthis plane always lie between two vortices in view from the side. The offset of these vorticesdescribes the observed flapping during vortex shedding.The vortex shedding frequency is mainly measurable in the shear layer and is noticeable viathe flow field up to the base pressures at Sr h ≈ .
2. The strongest fluctuations in the shearlayer mainly occur with Sr h ≈ . m = 1 to m = 4) whose temporal amplitudes have a peak in the frequency domain Sr h ≈ .
22. The measured frequencies in the flow field seem somewhat higher at first glance.However, these frequencies originate from very broadband peaks.The effects of vortex shedding can be traced back to the motions of the barycenter of thebase pressure and in the flow volume, but above all in the shear layer. The spectra of thedimensionless, spatially averaged surface pressure gradient show frequency peaks similar tothose in the shear layer and in the POD modes of the flow field (cf. Figure 5). Thus, thedominant frequencies in the shear layer in the range Sr h ≈ . Sr h ≈ .
2, which is assigned to vortex shedding, isbroadband and not dominant in the spectra of the barycenter of motion of the base pressuredistribution. Broadband capability can be considered typical for turbulent flows. The weakdevelopment of the peaks can be explained by observations of vortex shedding in the flow field.This weak response is due to relatively small vortices far away from the base, which isintensified by the diffuser. Vortices act as pressure sinks on surfaces. The closer a vortex is to the surface or the larger the vortex, the lower the pressure on the surface. The basepressure distributions show the lowest pressures in the lower section. This correlates with theobservation from the temporally averaged flow field, where in this area the vortex ring is verypronounced and close to the base. However, the vortices of vortex shedding are comparablysmall and far away from the base. This is because vortex separation mainly takes place in theupper dead water and close to the saddle point. Thus these structures are located far from thebase surface compared to the position of the lower part of the vortex ring or the length of thedead water area. In addition, the structures observed are smaller than half the height of thebase surface. The coherent structures of other flows, such as the K´arm´an Vortex Street in atwo-dimensional flow, have a much stronger effect on the base. In the wake of the SAE model,the size and strength of the vortices is much smaller and the distance to the base is larger. It isobvious that these properties of the size of the detached vortices and the distance to the baseis even further increased with the diffuser by the wake topology.With other three-dimensional bluff body flows and squareback vehicle models, the vortexseparation is also meander-shaped. However, the meander-shaped detachment of the vorticesis not limited to a part of the dead water, as is assumed in the SAE model due to the influenceof the diffuser. A meander-shaped vortex separation was proposed and documented for manyother three-dimensional bluff body flows, such as the circular or elliptical disc (Kiya and Abe,1999b; Yang et al., 2015), the sphere (Berger et al., 1990) and squareback vehicle models(Grandemange et al., 2013b; Duell and George, 1999). The frequencies of the vortex sheddingare all in the range Sr = 0 . ... .
22, in which the vortex shedding frequency is also for theSAE squareback model. Compared to the vortex shedding of the circular or elliptical disc(Kiya and Abe, 1999b; Yang et al., 2015) and to the documented squareback vehicle models,without diffuser (Grandemange et al., 2013b; Duell and George, 1999), the affected area issignificantly smaller in relation to the cross sectional area. In addition, the coherent structuresof vortex shedding move alternately in all directions in the wake of the circular disc (Yanget al., 2015) and of the squareback vehicle model (Duell and George, 1999). In contrast, thecoherent structures of vortex shedding on the SAE model only move away from the ground inthe vertical axis. As a special feature of the flow around an SAE squareback model and unlikethe behavior of the Ahmed squareback model (Grandemange et al., 2013b) it is shown thatalmost exclusively the upper vortex is involved in the vortex shedding. The area of the lowerpart of the vortex ring remains relatively stable. It is obvious that the influence of the diffuseron the velocity profiles in the wake is responsible for the stabilization of the lower area of thewake.
The meander-shaped vortex shedding in the upper half of the wake can be regarded as a basicform in connection with the vortex shedding on many squareback vehicle types. The geometryof the SAE model contains only the most outstanding features of a squareback vehicle. Thebehavior of the meander-shaped vortex shedding on the SAE model, which only takes place inthe upper half of the wake, can also be shown on the more complex DrivAer model geometry.For this purpose, the y/h = 0 plane in the wake of the 1:4 DrivAer model with a Reynoldsnumber of Re l = 3 . , was measured with PIV as on the SAE model. The basic vehiclegeometry is described by Heft et al. (2012) and the configuration of these measurements andsome details about the cooling air flow geometry by Kuthada et al. (2016). The flow fields weredecomposed with the POD. The first two modes also represent a convective pair of modes. oherent Structures in a Vehicle Wake 27 Figure 15 shows the reconstruction of this mode pair in four phase positions with streamlines.The representation is analogous to the reconstruction of the flow field in the plane on the SAEmodel (Figure 9). The rear end of the vehicle model can be seen on the left in each subframe. Itis noticeable that the saddle point is closer to the base or not as far from the rear edge as withthe SAE model. This is a feature that can be attributed to the continuous shape optimizationof the production vehicles that is taken into account in the DrivAer model. However, the wakeflow characteristics and the wake flow of the SAE model both have a greater curvature of thelower free streamline compared to the upper one. This provides a stronger upwards ambitionfor this streamline. Thus the basic shape and the arrangement of the vortex system in the deadwater is very similar to the SAE squareback model, although the base of the DrivAer model hasvery different degrees of surface curvature gradients. Streamlines thus show the lower vortex inalmost same position and unchanged size at all four phase positions. The upper, smaller vortexshows an increase and a downstream convection up to a detachment as with the SAE model.The vortex shedding of the DrivAer is therefore very similar to the SAE model. Together withthe fact that in all documented cases of three-dimensional bluff body flows a meander-shapedvortex shedding occurs, it is very likely that in the DrivAer model the vortex shedding is alsomeander-shaped in the upper half of the dead water. This leads to the hypothesis that thisvortex separation is basically valid for the majority of squareback geometries and possibly alsoSport Utility Vehicles (SUV).Figure 15: POD reconstruction of the wake flow on a DrivAer model in the y/h = 0 plane.
The low-frequency fluctuations of the wake represent a flapping that may interact with thevortex shedding. For the flow around three-dimensional bluff bodies, such as the circularor elliptical disc (Kiya and Abe, 1999b; Yang et al., 2015), the sphere (Berger et al., 1990)and squareback vehicle models (Grandemange et al., 2013b; Duell and George, 1999), a lowfrequency range of the following motion is documented parallel to the vortex shedding. Thespectra of the dimensionless, spatially averaged surface pressure gradient and the flow fieldmodes on the SAE model have very pronounced and dominant fluctuations in the low frequency range. These are also visible in the spectra of the flow field of the shear layer. In the spectraof the dimensionless, spatially averaged surface pressure gradient (Figure 5) two ranges ofaccumulations of fluctuations can be distinguished. These are the ranges Sr h ≈ .
07 and Sr h < .
06. The latter can only be seen in the dimensionless, spatially averaged surfacepressure gradient in y , i.e. in the motions of the barycenter of pressure in y -direction. This isto be interpreted as a deflection of the dead water mainly in the y -direction (i.e. the width),which takes place in very long time scales. When comparing the POD reconstruction of theflow field in the mid-frequency range (cf. Figure 11 and Figure 12), no vortex shedding can beseen when looking at the low frequencies. Rather, it is a lateral motion of the wake, combinedwith compression or stretching of opposite areas in the width (y-direction) and can therefore bedescribed as flapping. According to Brackston et al. (2016), the fluctuations at low frequenciesor long time scales correspond to the range to be assigned to the motions of the quasi-randomfluctuations and bi-stabilities. The distributions of the barycenter of pressure positions (cf.Figure 4) show that this is not a bi-stability. From the perspective of the spatial motionpattern, this flapping motion is comparable to the bi-stability of the Ahmed model, but withdifferent temporal behavior. The time scales of this motion are similar to flapping, as withthe circular disc (Yang et al., 2015) or the axial symmetric blunt body (Gentile et al., 2016b).Similar to Yang et al. (2015) describing the modulation of vortex shedding by flapping, the twocorresponding phenomena could interact with each other on the SAE model.The absence of bi-stability is in accordance with the findings of Grandemange et al. (2013a).Here, different combinations of aspect ratio and ground clearance of an Ahmed model wereexamined with regard to bi-stability. Measured against these parameters, the SAE squarebackmodel is just within the range of an occurrence of a bi-stability. However, it should be noticedthat the wake flow of the SAE model, especially through the diffuser, differs from that of theAhmed model. The diffuser causes the wake to move upwards. The flow topology is thereforerather comparable to that of the Ahmed model with an aspect ratio h/b = 0 .
75 and a groundclearance c/h = 0 . The detachment of a horseshoe vortex in the upper shear layer correlates well with the distri-bution of the base pressure fluctuations which have their maximum in the middle of the upperhalf. Therefore it is surprising at first that the shedding of the vortex is not better visible inchanges of the base pressure distributions. One possible cause is that the velocity fluctuationsof vortex shedding are poorly transferred to the base. Reasons could be the size, strength anddistance of the vortex compared to the base or the spatial phase shift of the vortex. Anothercause could also be the interaction of vortex shedding with the low-frequency motions.Some of the current research shows two processes that influence each other or may evenbe linked to each other (Rigas et al., 2017; Barros et al., 2017). In many flows of bluntbodies, low-frequency behavior was found (Grandemange et al., 2013b; Brackston et al., 2016;Duell and George, 1999; Gentile et al., 2016b; Rigas et al., 2014; Kiya and Abe, 1999b; Yanget al., 2015; Grandemange et al., 2012b,a). The low-frequency flapping motion observed in thiswork resembles the low-frequency flapping of axial symmetric blunt bodies or considering theamplitudes the bi-stability of some squareback vehicle models (Grandemange et al., 2013b,a).Many indications show that the vortex shedding interacts at least with the low-frequencybehavior. In some works (Kiya and Abe, 1999b; Yang et al., 2015) it is shown that the low- oherent Structures in a Vehicle Wake 29 frequency motion in the wake of axial symmetric blunt bodies causes a modulation of the vortexshedding. Furthermore, the work of Rigas et al. (2017) on the axial symmetric blunt body andof Grandemange et al. (2013b) on the Ahmed model show that the vortex shedding is alignedwith the low-frequency wake deflection.Another observation underlines that the interaction ofthe coherent structures of vortex shedding and low-frequency motions cannot be fully observedseparately. Rigas et al. (2014) investigated the coherent structures on the axial symmetricblunt body by the barycenter of pressure motion, POD and Fourier decomposition of thebase pressure distribution, whereby the vortex shedding could not be separated from the low-frequency motions in all methods. The findings of Gentile et al. (2016b,a), the POD and thebarycenter of pressure motion in wake planes lead to the same results in this respect.The findings in this work and the comparison with documented research show that thereis the possibility that the low-frequency modulation frequency may result from a dislocationduring vortex shedding as in the two-dimensional flow (Williamson, 1992). The motion of thebarycenter of pressure on the SAE model observed here is reminiscent of the constellation inthe two-dimensional flow. In the two-dimensional flow, the barycenter of pressure moves in thedirection perpendicular to the two-dimensional flow through the low-frequency dislocation. Atthe vertical axis of two-dimensional flow, the barycenter of pressure will move only by vortex-shedding (Bock, 2019). Likewise, the low-frequency motion of the barycenter of pressure on theSAE model mainly takes place in a direction perpendicular to the main direction of a motionof the barycenter of pressure as a result of the vortex shedding. Another analogy is that in thetwo-dimensional flow the low-frequency fluctuation in the base pressures is more easily detectedthan the vortex shedding. The low-frequency fluctuations are recorded at each measuring point.However, the fluctuations based on vortex shedding only become clearer by looking at certainmeasuring points or through certain filters (Bock, 2019). Finally, the results show anothercharacteristic that indicates a strong interaction or even combination of both processes. Sincethe vortex shedding on the SAE model is phase-shifted over the width, it is accompanied by aflapping motion. It is very likely that an additional (low-frequency) flapping motion changesthis phase shift and thus also acts as a dislocation. Nevertheless, the dislocation seems unlikelyor has not yet been determined as the cause for modulation on the basis of various studies(Kiya and Abe, 1999b; Yang et al., 2015).On the basis of this work and other publications, a clear statement cannot be made onwhether dislocation does or does not occur. A modulation of vortex shedding is very obviousas many authors conclude from their results. Dislocation, on the other hand, cannot be clearlydemonstrated. Further investigations would be necessary to prove this. For example, structuresof vortex shedding during shedding could be traced to detect dislocations if necessary.
In the present study, the coherent structures in the wake of a squareback SAE model were ex-amined experimentally and numerically. Time-resolved base pressures and velocities in differentplanes were measured statistically with pressure transducers and PIV. Using VLES simulations,additional time resolved and spatially coherent data were generated in the wake of the SAEmodel. Small-scale fluctuations were spatially filtered by observing barycenters of pressure,dimensionless, spatially averaged surface pressure gradients at the base and in the flow field byPOD. An additional differentiation of time scales was achieved by spectra.It has been shown that the shape of the vortex ring in the wake can differ on the differentsquareback vehicle models. These differences are probably geometry-dependent. This has an effect on the vortex shedding, on base pressure fluctuations and on the behavior of fluctuationsin the low frequency range. It can be assumed that these flow properties must be taken intoaccount in flow control measures that affect the behavior of the fluctuations. In addition, thepotential to reduce aerodynamic drag could also change due to the different behavior of theflow situation.The structures of vortex shedding are difficult to detect, especially in the base pressures ofthe SAE squareback model, which can nevertheless be used as feedback sensors by means ofsuitable filtering. The dead water of the SAE squareback model consists of a vortex ring whichis located close to the base in the lower part and is deformed by the diffuser in the upper part.The shape of the vortex ring and thus the dead water area differs from that of an Ahmedsquareback model. The vortex shedding on the SAE model can be interpreted as a meander-shaped process in the upper half of the wake with an average frequency of Sr h ≈ .
2. Due tothe position and the size of the vortex, the base pressures only slightly reflect the shedding ofthe vortex. This weak response is due to relatively small vortices far away from the base. As afeedback sensor for active flow control of vortex shedding, the base pressures are therefore notoptimal. Filtering the sensor signals via the barycenter of pressure can improve this problem.In the context of the investigations of Brackston et al. (2016) it can be seen that the signal ofthe barycenter of pressure is suitable to influence the vortex shedding at the Ahmed model in afeedback loop, even though it was excited and not reduced by this example and thus probablythe aerodynamic drag was increased.The meander-shaped vortex shedding in the upper half of the wake can be regarded as a basicform in connection with the vortex shedding on squareback vehicle types. This is confirmed bythe meander-shaped vortex shedding in other three-dimensional blunt bodies and squarebackvehicle models (Grandemange et al., 2013b; Barros et al., 2017). However, the meander-shapedshedding of the vortices is in these cases, in contrast to the observed vortex shedding of theSAE squareback model not limited to a part of the dead water. This distinction is attributedto the influence of the diffuser.The low-frequency motions of the wake represent a flapping in width direction. It is suspectedthat the flapping may interact with the vortex shedding. The lack of bi-stability is in accordancewith the findings of Grandemange et al. (2013a). Separation of low frequency, or detection ofdislocations remains to be investigated in further work.The exact form of vortex sheddings could be relevant for the positions of an active flowcontrol measure. The findings of this study show that there are strong influences of geometryon the fluctuation motion in the low frequency range, but also on the vortex shedding. Theunderstanding of the vortex shedding process on the SAE model and on squareback vehicletypes suggests the response to active control. The shedding of the vortex loop and the reactionto a control approach should be basically very similar in the axial symmetric body (Rigas et al.,2017) and the Ahmed model (Barros et al., 2017). Barros et al. (2017) showed a correlationof the sensitivity of vortex shedding to the flow control position with the preferred directionof vortex shedding. In the vortex shedding of the SAE model, the activity range is limited tothe upper half of the wake compared to these flows. For such an application, only a reducedrange of pulsed or synthetic jets would have to be excited (possibly only on the side and thetop edges). At the same frequencies and amplitudes, similar results can be expected as forBarros et al. (2017) and Rigas et al. (2017). Up to now, however, in doing so aerodynamic draghas been increased and the energy content of the fluctuating motions of the low frequenciesreduced. Nevertheless, the findings of Barros et al. (2017) show that a relevant reduction ofthe aerodynamic drag forces is possible by control approaches on the vortex shedding. oherent Structures in a Vehicle Wake 31
Another consequence of the results of this study is that the form of vortex shedding couldchange due to passive or active control approaches of the flow field. The findings of vortexshedding on the SAE model compared to the Ahmed model show the possible influence ofgeometry on coherent structures. This also means that an influence on the temporally averagedflow in the wake, e.g. by any means of aeroshaping, could severly change the process of vortexshedding again.Further detailed studies on the coherent structures of squareback vehicle models or three-dimensional blunt bodies with different geometries could help to clarify the following questionsregarding the impact on aerodynamic forces. What is the effect of the fluctuations and how highare the energy components of the coherent structures? In applications this would be relevantfor the influence on aerodynamic drag, but also on other forces, e.g. in driving dynamics. Thiscould point the way to show how high the energy input of an active control measure could be, sothat the energy saved is still worthwhile in optimising the behavior of a coherent structure. Thepresent study suggests that this is to be expected with significantly lower energy proportionscompared to the two-dimensional flow during vortex shedding.A flow control measure, such as the place of excitation in active flow control, must dependon where the processes take place and which is the main direction of motion. The presentinvestigations have shown that this can, however, depend on the geometry. A more systematicinvestigation of geometric influences such as different diffuser angles, rear shape and possiblyothers could provide further important information here.In conclusion, these results improve the understanding of the behavior of coherent structuresand the effect on base pressure, i.e. vehicle forces for squareback vehicle types. This will becrucial in developing new strategies to influence and control vehicle forces.The author acknowledges the support of the IVK at the University of Stuttgart where theexperiments and simulations were performed. Furthermore, the support of the Friedrich-und-Elisabeth-BOYSEN-Stiftung is thankfully acknowledged for the funding under grant BOY11-No.77. The author also warmly thanks Jochen Wiedemann, Nils Widdecke, Timo Kuthada,Christoph Sch¨onleber, Alexander Hennig, Max Tanneberger and Daniel Stoll for fruitful dis-cussions.
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