Cavitating Flow Behind a Backward Facing Step
CCavitating Flow Behind a Backward Facing Step
Anubhav Bhatt, Harish Ganesh, S.L. Ceccio
University Of Michigan, Ann Arbor
Abstract
Flow topology and unsteady cavitation dynamics in the wake of a backwardfacing step was investigated using high speed videography and time-resolvedX-Ray densitometry, along with static and dynamic pressure measurements.The measurements are used to inform the understanding of underlying mech-anisms of observed flow dynamics as they are related to shock wave inducedinstabilities. The differences in cavity topology and behaviour at different cav-itation numbers are examined to highlight flow features such as the pair ofcavitating spanwise vortices in the shear layer and propagating bubbly shockfront. The shock speeds at different cavitation conditions are estimated basedon void-fraction measurements using the X-T diagram and compared to thosecomputed using the one-dimensional Rankine Hugoniot jump condition. Thetwo speeds(computed and measured) are found to be comparable. The effectof compressibility of the bubbly mixture is determined by estimating the localspeed of sound using homogeneous ‘frozen model’ and homogeneous ‘equilib-rium model’. The current estimation of sound speed suggests that expressionused to determine the local speed of sound is closer to the classic ‘frozen model’than the ‘equilibrium model’ of speed of sound.
Keywords:
Shockwaves, speed of sound, X-Ray densitometry
1. Introduction
Developed cavitation in reattaching shear flows can result in unwanted noise,erosion and vibration various hydrodynamic applications. Despite being exten-sively studied and reported, the complexity of these turbulent, high void fraction
Preprint submitted to International Journal of Multiphase Flows December 8, 2020 a r X i v : . [ phy s i c s . f l u - dyn ] D ec ow makes direct interrogation quite challenging. Given the geometric simplic-ity and the availability of previous results detailing the single phase phenomena,the flow behind a backward facing step is an ideal candidate for experimentalinvestigation of the effect of cavitation on turbulent shear layers.A schematic of the single-phase flow topology downstream of the backwardfacing step is shown in Figure 1. The flow has a sharp separation point, fixedto the edge of the step. As the single phase flow separates from the edge, thefree shear flow rolls up to create large scale spanwise vortices, similar to those ofmixing free shear layer structures (Troutt et al. (1984)). This shear flow will ulti-mately reattach to form a recirculation bubble underneath the shear layer. Fur-ther downstream the wall effects start to dominate (Bradshaw & Wong (1972))as shear layer curves sharply downward in the reattachment region and impingeson the wall. In the reattachment region, the shear layer is subjected to distort-ing forces due to effects of stabilizing curvature, adverse pressure gradient andstrong interaction with the wall (Eaton & Johnston (1981)). Downstream of thereattachment a new sub-boundary layer begins to form. It can be clearly seenthat this geometrically simple flow has almost all the features of separating andreattaching shear flows including a mixing layer, recirculation, re-attachmentand a redeveloping boundary layer. In addition, for backward facing step flowconfigurations, the boundary of the flow above the step may be in close prox-imity to the step to influence the step flow, making the expansion ratio also aparameter of interest (Biswas et al. (2004)).Bernal & Roshko (1986) developed a plane mixing-layer model highlightingthe streamwise vortex structures and noted structures of dominant spanwisevortices.They reported the ‘braid’ region between adjacent spanwise vortices tocontain an array of streamwise vortices which snake between neighboring eddies.The spanwise vortex structures shed near the separation point can be subjectedto vortex pairing interaction upstream of reattachment region, indicating thatthe growth of the separated layer may be strongly dependent on the pairingprocess (Troutt et al. (1984)). Using PIV measurements and pattern recognitionScarano et al. (1999) presented details of the flow structures observed within2he shear layer.From a dynamics perspective, a low frequency “flapping motion” in thewhole separated region has been observed (Eaton & Johnston (1981)). Onepossible explanation for this motion is attributed to the mass imbalance betweenshear layer entrainment from recirculation region and re-entrant flow near thereattachment point(Eaton & Johnston (1981)). This imbalance is said to becaused by short term breakdown of spanwise vortices in shear layer that leadsto decrease in the entrainment rate, causing a change in volume of recirculatingfluid. Another explanation is attributed to a reduction in the amount of massengulfed by the separation bubble (Driver et al. (1987)) causing the bubble tomomentarily collapse. Yet another explanation of flapping motion states thatthe re-attachment point is unsteady and it moves upstream intermittently inbursts, that splits shear layer into two halves and compresses the fluid beneaththe shear layer (Hasan (1992)). This compressed fluid is subsequently ejected,pushing shear layer outward and causing self-sustained flapping. The flappingmotion occurs at a non-dimensional frequency of O(1) (Spazzini et al. (2001),Wee et al. (2004)).Liquid flows in the wake of the step can experience cavitation if the localpressure in the shear layer falls below the vapor pressure. For turbulent shearflows, cavitation inception(defined as the first occurrence of observable cavita-tion) is known to first occur in streamwise braids as they are stretched betweenthe spanwise vortices of the shear layer (Katz & O’hern (1986) ; O’hern (1990)).At this stage, cavitation bubbles are hypothesized to decouple the stretching androtation rates of streamwise vortices (Belahadji et al. (1995)) and known to notsignificantly alter bulk flow topology (Iyer & Ceccio (2002)).As the extent of the cavitation increases, it can influence the vorticity inthe shear layer, and gas can begin to fill the separated region of the flow. Forcavitating wakes, the presence of developed cavitation has a significant impacton large scale structure dynamics like vortex spacing, shedding frequency andconvective speeds of spanwise structures (Franc & Michel (1983); Young & Holl(1966); Aeschlimann et al. (2011); Wu (2019)).3 igure 1: Topology of the non-cavitating flow behind a backward facing step. Classical re-entrant jet driven shedding and propagating bubbly shock-wavedriven shedding are two known mechanisms that govern shedding dynamics indeveloped partial cavities (Ganesh et al. (2016); Wu et al. (2019)). The re-entrant liquid flows are primarily driven by adverse pressure gradient at thecavity closure (Callenaere et al. (2001)). The bubbly shockwaves on the otherhand are known to occur due to the effect of compressibility of the vapor-liquidmixture that reduces the local speed of sound, thereby driving flow to go locallysupersonic. These features have also been observed in separated cavity flowsformed behind obstacles (Barbaca et al. (2019)). A proper expression for thespeed of sound is necessary to accurately estimate the underlying dynamics ofa rich bubbly shock, thereby explaining shock-wave induced instabilities thatmay effect cavity shedding.In the current study, we will examine the flow features of developed cavitiesformed in the wake of a backward facing step, and quantitatively understand theunderlying cavity dynamics through high speed videography and time resolvedX-ray densitometry. Our goal is to both gain a better understanding of theflow physics as well as to develop a data set that can be used for the validationof computational models. The manuscript is divided into five parts. First,we present the experimental setup; we then define three flow regimes of interest(Type A, B and C) and describe their flow dynamics (cavity shedding frequencyand pressure inside cavity). Finally we relate these dynamics to the underlyingeffect of flow compressibility. 4 . Experimental Setup
Experiments were carried out in the Michigan 9” recirculating cavitationchannel. The upper leg of the tunnel has 6:1 contraction leading to 228.6 mmdiameter section that leads to 209.6 mm by 209.6 mm square section with filletedcorners. To minimize the baseline attenuation in X-ray densitometry for non-cavitating flow, the channel is further reduced to 78.4 mm by 78.4 mm testsection. A fifth order polynomial profile (f(x)) leads to the flat ramp of thebackward facing step. Step height (H) was chosen to match the blockage usedby researchers at Johns Hopkins( ? ). The test section profile is shown in Figure2. The inlet velocity ( u ) was measured by measuring the pressure drop acrossthe 6:1 contraction using Setra 230, 0-68 kPa, differential pressure transducerwith an accuracy of 0.25% of full scale (FS). From this measured u velocity atstep( u H ) is determined using mass conservation( h ∗ h ∗ u = ( h − H ) ∗ h ∗ u H ) Forcurrent dataset u H of 8, 10 and 12 m/s are maintained for Re H = 8 . · , 10 . · and 12 . · respectively to operate the water tunnel within practical inletpressures ( p ). Inlet pressure ( p ) was varied from inception to before super-cavitation for a σ range of 0.5 to 1.0. p was measured using Omega PX409-030A5V, 0-206 kPa static pressure transducer (0.08 % FS accuracy). Pressure p was measured using differential pressure transducer Omega PX2300, 0-34kPa (0.25 % FS accuracy) between pressure taps at p and p . Omega PX409-005AI-EH, 0-34 kPa (0.08 % FS accuracy) static pressure transducer measurespressure at X = 1.5H ( p ); while Omega PX203, 0-206 kPa (0.25 % FS accuracy)static pressure transducer measures pressure at X = 8H ( p ). Dynamic pressuretransducer PCB 113M231, 7.04 µA /kPa sensitivity is used to measure dynamicpressure dp . The signal is acquired at a sampling frequency of 50 kHz for 3seconds. Finally the overall pressure drop across the test section is measuredusing Omega RX2300, 0-34 kPa (0.25 % FS accuracy). All pressure taps onoptically accessible acrylic windows have 1.6 mm diameter. All the staticand dynamic pressure measurement locations are summarized in Table 1. The5 igure 2: Schematic of the backward facing step mounted in the water tunnel. Top and sidefield of view configurations for the high speed videography are shown. For X-ray densitometrya 115 mm by 28 mm view, roughly similar to the highlighted side view is taken (see Figure 5) .dissolved oxygen was maintained below 20 % of saturation for all experiments.2-D planar PIV was done to characterize the single phase flow of the setup bydetermining the boundary layer thickness at the step and reattachment length.High speed cinematography (HS) was performed using time synchronized Phan-tom 710 cameras at 5000 fps for 1.5 s with both top and east side view. Cine-matographic X-ray densitometry system (Ganesh et al. (2016), ? ) was used tomake spanwise-averaged quantitative void fraction ( α ) measurements at 1000fps for 0.787 s exposure time and 0.125 mm spatial resolution.
3. Results
The experiments were conducted by keeping the velocity at step u H , fixedat 8, 10 or 12 m/s and reducing inlet pressure ( p ) from inception (near σ = 1)to near super-cavitation (near σ = 0 . p further,greater extent of cavitation occurs primarily between X = -3H to -5H, classifiedas Type A or ‘developing cavitation’ phase of the flow. Developing cavitationis characterized by the presence of vapor bubbles in spanwise vortices of theshear layer. A high-speed video snapshot of developing cavitation is depicted inFigure 4(a). 6 able 1: Inlet parameters, transducer locations and geometry of the step.Figure 3: Cavitation inception in the streamwise direction is highlighted (streamwise ’braids’Katz & O’hern (1986)). Intermittent cavitation occurs at σ = 0 .
98 for u H = 10 m/s. Thered lines highlight the edge of the step. p , unsteady cavitation designated asType B is observed. Type B self-sustained shedding cavitation is the regimewhere periodic oscillation of the cavity length occurs at comparatively higherfrequency (Typical cavity fluctuations for Type A occur below 10Hz). As p isdropped even further, Type C cavitation is observed. This is the stage rightbefore super-cavitation where shedding still occurs albeit at a much lower rateand cavity is filled with vapor. The instantaneous cavity lengths for Type C startto extend beyond the field of view of the X-ray densitometry system (roughlythe side view shown in Figure 2). The three phases of cavitation are shown inFigure 4 and Figure 5. Figure 4: High speed snapshots of three different cavitation regimes (a, b, c for σ =0.78, 0.72,0.60 respectively) at u H = 10 m/s. Type A comprises of cavitation in shear layer mostly nearreattachment region(away from separation point). Types B and C shedding regimes are ofprimary interest in this study. For inception and developing cavitation regimes the measured mean void8 igure 5: The mean void fraction fields (snapshots) of the three cavitation regimes (a, b, cfor σ =0.78, 0.72, 0.60 respectively) at u H = 10 m/s. The mean length of cavity < L C > /H and distance to maximum void fraction value (in the mean field, < L M > /H are shown. Theactual densitometry data captures entire 0-100 % void fraction scale, but color scale artificiallylimited to 0-2% for Type A and 0-25% for the rest. fraction values are less than 10%. The X-ray densitometry system yields 2%uncertainty in the reported void fraction values. This is the main reason whyhigher void fraction Type-B and Type-C regimes are of primary interest in thecurrent study (acceptable signal to noise ratio (SNR)). We use the mean void fraction flow fields to determine the downstreamlocation where void fraction values ( α MEAN ) are 0.05 in the region of cavityclosure, defining the mean cavity length < L C > . The mean void fraction field9 igure 6: Mean cavity length( < L c > ) and location of maximum void fraction( < L M > ) for u H = 8 m/s ( (cid:4) , (cid:3) ), 10 m/s ( (cid:78) , (cid:52) ), 12 m/s ( ◦ , • ) for different σ . Closed symbols are for < L c > and open symbols for < L M > . is estimated as : α MEAN = (cid:88) N =0 α ( x, y, N )787 (1)The distance to the maximum void fraction ( max ( α MEAN )) in the field isrecorded as < L M > . Both these distances are measured from the separationpoint on the step, as shown in Figure 5(b) and plotted in Figure 6.It is observed that higher the Re, for a given σ , the lower the mean cavitylength. Conversely, L M lies between 3 to 4H, regardless of the Re. This can bejuxtaposed with the shear stress distribution reported in the previous literature(Jovic & Driver (1995)). The shear stress maximizes roughly at 0.67 reattach-ment length, which is 3.7H for their specific single phase flow configuration.10 .2. Typical Shedding Cycle To better identify specific flow features during the shedding cycle observedin Type B and Type C cavities, high speed cinematographic frames of the floware shown in Figure 7. The primary cavitating spanwise vortex (green) andsecondary vortex (blue) are seen interacting in Figure 7(c). This interaction canbe seen through streamwise vortex bridges formed between the two spanwisevortices. Both these vortices have the same sense of rotation(anticlockwise).During the filling period (Figure 7(f)-(g)) the side-view of the cavity appearsquasi-steady. The ejected spanwise vortex at the end of a shedding cycle can bevisualized in Figure 7(l).The X-Ray densitometry image frames of a typical Type-B shedding cycleare shown in Figure 8. The cycle begins with cavitation in the core of a spanwisevortex (vortex 1), highlighted in green between X = -0.5H to X = -2H (Figure8(a)). As this structure convects downstream, it grows in size and interacts withanother vortex (vortex 2, highlighted in blue). Subsequently the vortex pairbecomes indistinguishable and a filled cavity starts to emerge between X = -2Hto X = -6H seen in Figure 8(d)-(f). The seemingly quasi-steady snapshots shownin Figure 7(f)-(g) can be compared to Figure 8(e)-(f) to indicate additional flowfeatures captured through densitometry. The cavity consistently grows from theaft, i.e. it fills from roughly around the reattachment zone until a filled cavitystate is achieved.Once a fully filled cavity is formed and reaches a high instantaneous voidfraction value( α = 0 . − . igure 7: The timeseries of high speed snapshots of a periodic Type B shedding cycle. Caseat σ =0.72 for u H = 10 m/s. Cycle starts with visible cavitation in spanwise vortex(b). Therecirculation bubble and shear layer continue to fill with vapor(e)-(g), until an adverse flowfront(i) causes the cavity to disappear(shed cavity in (k)). igure 8: X-Ray timeseries of a typical Type B shedding cycle at σ =0.72 for u H = 10 m/s.The void fraction ( α ( x, y, t )) range is set to 0-70%. The adverse flow front highlighted in (h)resembles the bubbly shock front igure 9: X-Ray timeseries of a typical Type C shedding cycle at σ =0.60 for u H = 10 m/s. The void fraction ( α ( x, y, t )) range is set to 0-85%. Compared to Type B, Type C cavitiesare longer, have higher vapor content and collapse at a slower rate. High speed snapshots ofType C appear quasi steady, unlike the X-Ray timeseries which clearly show shedding. igure 10: Mean void fraction contours at u H = (a) 8 m/s, (b) 10 m/s and (c) 12 m/s at σ = 0 .
72. 5 slices at
X/H = -1, -3, -4, -5, -7 are taken to compare void fraction values acrossthe three velocities shown in Figures 11,12, 13.
Based upon local pressure distribution and flow dynamics, void fraction val-ues can vary significantly. The mean void fraction profiles for the onset of TypeB cavitation at different Re are shown in Figure 10. It can be seen that themean void fraction profiles for the onset of shedding are almost identical acrossdifferent Re.X-slices of the mean void fraction field for Type B shedding regime at X = -H, -3H, -4H, -5H and -7H are shown in Figure 11. The mean void fraction valuesare comparable across different Re. X-slices slices for the Type C shedding areshown in Figure 12. The mean void fraction values, now higher in magnitudeare still similar across Re. Taking similar slices on the Root Mean StandardDeviation (RMSD) void fraction field yields comparable RMSD values acrossthe three Re for Type B regime in Figure 13(a). The RMSD field is generated15 igure 11: Mean void fraction values(2% uncertainty) along slices Y/H = -1 to 1.5; u H =8 m/s ( (cid:3) ), 10 m/s ( (cid:52) ) and 12 m/s ( ◦ ) at σ = 0 .
72 (Type B). The mean void fraction valuesare comparable across three Re.Figure 12: Mean void fraction values for Type C; u H = 8 m/s ( (cid:3) ), 10 m/s ( (cid:52) ) and 12 m/s( ◦ ) at σ = 0 .
60. No significant change in void fraction values due to Re is seen.
16s follows : α RMSD = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:80) N =0 ( α ( x, y, N ) − α MEAN )
787 (2)Comparatively larger RMSD values are seen in Type C shedding regime (Figure13(b)). With decreasing cavitation number, void fraction fluctuations increase.As mentioned earlier, the reported void-fraction values have 2% uncertainty inmeasurements.
The cavity shedding cycle highlighted in Figure 7 and Figure 8 occurs pe-riodically, and as the flow transitions from Type A all the way to Type C, theshedding frequency, f, also changes. The Strouhal number (St) used to non-dimensionalize the shedding frequency is given by St = f · u H The shedding frequency is determined from two time varying signals. First,the spatially averaged local void fraction averaged over a 1.25mm by 0.875mmprobe area located at (-5.5H, -0.5H). The shedding frequency observed throughthe void fraction signal does not change, for the probe located anywhere be-tween X = -6.5H to -0.5H (Y= -H to 0). Second, the shedding frequency isrecorded with the dynamic pressure transducer ( dp ). The dynamic pressuresignal( dp ), the time varying void fraction signal and corresponding power spec-tral densities for a Type B and Type C case are shown in Figure 14 and Figure15 respectively. The primary shedding frequencies lie between 10-70 Hz. Theresulting Strouhal number is defined with the strongest frequency peak, and itmonotonically decreases with decreasing σ , shown in Figure 16. Note that Stis not strongly influenced by changes in Re. The static wall pressure is measured at different streamwise locations be-neath the cavity. The inlet pressure p is used as the basis to find non-dimensional coefficient of pressure Cp i = p i – p · ρ · u where i=1, 2, 3 correspondto the pressure measured at the three static pressure taps beneath the cavity .17 igure 13: RMSD values for the three Re; u H = 8 m/s ( (cid:3) ), 10 m/s ( (cid:52) ) and 12 m/s ( ◦ );(a)Type B;(b) Type C. RMSD values for Type B are comparable, meanwhile for Type C, RMSDof cavity void fraction is sensitive to Re. This can be partially attributed to different meanpressures at these Re that affect local pressure fluctuations. igure 14: Dynamic pressure signal(a, · · · ) synchronised with time varying void fractionsignal(b, − ) and corresponding power spectral density(c) for a Type B cavitating case at σ =0.73 at u H = 10 m/s. The highest energy peaks are close. igure 15: Dynamic pressure signal(a, · · · ) synchronised with time varying void fractionsignal(b, − ) and corresponding power spectral density for a Type C cavitating case at σ =0.59 at u H = 10 m/s. Compared to Type B we can see a lower shedding frequency and lowerpressures dp . igure 16: The change in strouhal number(shedding frequency) with decreasing σ . u H = 8m/s ( (cid:3)(cid:4) ), 10 m/s ( (cid:52) (cid:78) ) and 12 m/s ( ◦• ). The strouhal numbers obtained through the voidfraction signals( α ) are shown in filled symbols, while open symbols depict strouhal numberfrom dp signal.Figure 17: The values of − Cp (filled, (cid:4)(cid:78) • ) and − Cp (open, (cid:3) (cid:52)◦ ) as σ is reduced acrossthree Re; u H = 8 m/s ( (cid:3)(cid:4) ), 10 m/s ( (cid:52) (cid:78) ) and 12 m/s ( ◦• ). Magnitude of Cp is higher than Cp for all three Re indicating that behind the step, the pressure drops between pressure tapsat location 1 and 2. igure 18: σ c with decreasing σ (Closer look at - Cp ). u H = 8 m/s ( (cid:4) ), 10 m/s ( (cid:78) ),12 m/s ( • ). At lower cavitation numbers, pressure inside the cavity approaches vapor pressure. The Cp values at different σ for p and p are shown in Figure 17. − Cp values are higher than − Cp values across all three Re. Both Cp and Cp valuespeak around σ = 0.71-0.78, which corresponds to onset of Type B shedding.As cavity fills with vapour ( σ is reduced), the − Cp value is seen to decrease.We can define a cavitation number based on cavity pressure as ( σ c ; where σ c = − Cp . In Figure 18 we see that the cavity pressure approaches vapor pressure(–) as σ is reduced. This means that even during Type B and Type C sheddingcycles, the cavity is not homogeneously filled with vapor, as also evidenced bythe void fraction measurements.-Cp for p and p are shown in Figure 19. Cp remains comparativelyunchanged across different cavitation regimes. At high cavitation numbers Cp is negative, indicative of higher pressure in the re-attachment region comparedto the inlet pressure as suggested in previous literature(Jovic & Driver (1995), ? ). − Cp increases drastically as flow transitions from Type B to Type C. Thisis due to drop in p as the cavity grows larger ( p approaches vapour pressure).22 igure 19: The values of − Cp ( (cid:4)(cid:78) • ) and − Cp ( (cid:3) (cid:52)◦ ) as σ is dropped across three Re; u H = 8 m/s ( (cid:3)(cid:4) ), 10 m/s ( (cid:52) (cid:78) ) and 12 m/s ( ◦• ) The flow front shown in Figure 8(g) primarily drives the shedding(collapse)of the cavity, and this flow front is analogous to the propagating shockwaves inrich bubbly mixtures reported in previous literature (Ganesh et al. (2016)). As σ is reduced (flow transitions from Type A to B) the vapor content in shearlayer and recirculation bubble increases. The bulk modulus of the vapor liquidmixture decreases with decreasing σ leading to higher compressibility in theshear layer and the recirculation bubble. Also with increasing vapor content thelocal speed of sound in the mixture decreases (Brennen & Brennen (2005)), andat optimal void fraction values the flow becomes locally supersonic. This makesthe flow susceptible to formation of bubbly shock fronts that can govern overallthe cavity shedding mechanism.As the cavitating spanwise vortices (highlighted in Figure 8(c),(d)) and sub-sequent bubbly shock fronts propagate over the differential pressure transducer( dp ) corresponding pressure rise can be recorded. This is shown in Figure 20,which highlights the different pressure rises over the dynamic pressure trans-23ucer ( dp ) and the local void fraction probe.Between stages I and II, a pressure rise is recorded, with the differentialpressure dp rising with the passage of the front. When the flow front impingeson the step, another larger pressure pulse is created (III). The collapse of theshed vortices cause the remaining pressure spikes (IV and V).The speed at which this flow front propagates ( U RH ) can be predicted bythe one dimensional Rankine-Hugoniot jump condition found by simple massand momentum conservation across a 1-D shockwave in the shock-wave frameof reference: U RH = dρρ L (cid:20) (1 − α )(1 − α ) ∗ ( α − α ) (cid:21) (3)where, ρ L =density of liquid(water) and α , α are void fraction values preand post shock. Since we have measured these values, we can compare thepredicted and measured front propagation speed. The α and α values arefound by spatially averaging void fraction values in void fraction probe as thefront convects over the dynamic pressure transducer, creating the correspondingchange in pressure, dp .The propagation speed U s is independently measured using the X-T dia-grams of near wall void fraction, as shown in Figure 21. Comparing the esti-mated Us from X-T diagrams and U RH computed using Figure 3 shows a verygood agreement. The compressibility of the vapour-liquid mixture affects the periodic shed-ding of the cavity, evidenced by the formation of the propagating bubbly shockfronts. Quantifying the local speed of sound of the bubbly mixture helps eluci-date the tendency of the flow to become supersonic at a given σ .Due to complex nature of the rich bubbly mixtures seen in type B and typeC flows, and absence of local velocity/pressure fields it is difficult to make pre-cise experimental measurement of the local speed of sound. Instead we canestimate the local speed of sound by using simplified expressions suggested in24 igure 20: The differential pressure ( dp ) and corresponding X-ray densitometry snapshots asthe shock-front moves past the transducer and hits the step during a shedding cycle. σ =0.72for u H = 10 m/s. igure 21: X-T diagram at Y/H = -0.25; σ =0.72 for u H = 10 m/s. The shock propagationis assumed to be linear and inverse of the slope of X-T diagram gives shockspeed.Figure 22: Comparison between computed shock-speed, U RH ( (cid:3) ) and shock-speed estimatedfrom X-T diagrams, U S ( ◦ ) for varying σ shows a good agreement between U S and U RH . (cid:15) L fraction of the liquid phase exists in thermodynamic equilibrium, then (1 − (cid:15) L )fraction will be insulated (similarly for (cid:15) V fraction of the vapour phase). Thegeneral qualitative expression that can be used for both the models to estimatethe speed of sound ( c i ) is given by:1 ρ m c i = αp i [(1 − (cid:15) V ) f V + (cid:15) V g V ] + 1 − αp i η (cid:15) L g ∗ p cη (4)where, i=1,2 & 3, corresponds to the pressure measured at the three staticpressure taps beneath the cavity. Mixture density ρ m = αρ V + (1 − α ) ρ L .Quantities f and g define thermodynamic properties of respective phase. Forwater g ∗ = 1.67, η = 0.73, g V = 0.91 and f V = 0.73 (Brennen & Brennen (2005)).The pressure p c ( = 22.06 MPa) is the critical pressure of water. (cid:15) V = (cid:15) L = 1corresponds to ‘homogeneous equilibrium model’ while (cid:15) V = (cid:15) L = 0 gives the‘homogeneous frozen model’ from Figure 4. The Mach number is then calculatedusing u H as: M i = u H c i (5)Figure 23 highlights that the experimentally observed onset of shockwaves(Type B shedding regime shown in Figure 8) approximately coincides with thequalitative estimation of sound-speed using the ‘homogeneous frozen model’.The flow is seen to transition from subsonic regime at Type A to supersonic27 igure 23: M i estimated using the ‘homogeneous equilibrium model’ ( (cid:70)(cid:7) (cid:73) ) and ‘homoge-neous frozen model’ ( (cid:70) (cid:70) ♦(cid:66) ) for speed of sound; M ( (cid:70)(cid:70) (cid:70) ) M ( (cid:7)♦ ) & M ( (cid:73) (cid:66) ). u H = 10 m/s .( − ) depicts M = 1 and indicates when the flow becomes supersonic ( M >
1) at the onset of Type B regime for M and M . M being subsonic( < u H instead of the local mean velocitynear p (mean flow velocity at p is not directly measured). The ‘homogeneousequilibrium model’ is seen to consistently overestimate M i and does not capturethe subsonic to supersonic transition suggested by the formation of shock fronts(Type A to Type B transition region). Thus, this suggests that the actual speedof sound is closer to that given by the ‘homogeneous frozen model’.The Mach number values estimated using the ‘homogenous frozen model’ forthe three Re are shown in Figure 24. It is seen that M i decreases monotonicallywith increasing σ . M , M and M do not have a clear Re dependency.
4. Conclusions
In the current study on a backward facing step we have identified threecavitating regimes observable under the x-ray densitometry and high speed cin-28 igure 24: The variation of M i with decreasing σ across three different Re esti-mated using ‘homogeneous frozen model’; u H = 8 m/s ( (cid:3)(cid:3)(cid:3) ) , m/s ( (cid:52)(cid:52)(cid:52) ) & 12 m/s ( ◦ ◦◦ ); M ( (cid:3) (cid:52)◦ ) , M ( (cid:3) (cid:52)◦ ) & M ( (cid:3) (cid:52)◦ ). (-) depicts M = 1. ematographic system. Throughout the study we determine the effect of changein Re on various cavity properties.Mean cavity lengths have a Re dependency as seen in Figure 6. During ashedding cycle the stronger spanwise vortices incept and grow in size as theyconvect away from the separation point. The vortex-pair interaction dominatesthe cavity filling part of the shedding cycle as visualized in Figure 7. An ad-verse flow front (shockwave) moving towards the step eventually diminishes thecavity (Figure 8(g)-(i)). The shedding frequency decreases monotonically withdecreasing σ and does not have Re dependence. The mean void fraction valuesin the cavity are comparable across the three Re.Static pressure p , p and p inside/around the cavity are measured as high-lighted in Table 1. The local pressure first drops and then recovers between p and p . Looking at Cp in Figure 18 shows that the cavity is not homogenouslyfilled with vapour. As σ is dropped (Type C and below) p is seen to approach29apour pressure.The dynamic pressure ( dp ) gives insight into the pressure fluctuations in-duced due to convecting shock-fronts. The shock speed computed using a simple1-D Rankine Hugoniot jump condition lines well with the shock-speed estimatedthrough the X-T diagram. The simple 1D assumption holds true even for shock-waves in the current complex flow system.Finally a simplified expression is used to qualitatively estimate the localspeed of sound and corresponding Mach number are discussed. The ‘homoge-neous frozen model’ appears to be consistent with the observed formation ofshock fronts with decreasing cavitation number.
5. Acknowledgements
Current study has been funded by Office of Naval Research as a part of Mul-tidisciplinary University Research Initiative(MURI) titled ‘Predicting turbulentmulti-phase flows with high fidelity’ (Grant: N00014-17-1-2676) under programmanager Dr. Ki-Han Kim.
References
Aeschlimann, V., Barre, S., & Legoupil, S. (2011). X-ray attenuation measure-ments in a cavitating mixing layer for instantaneous two-dimensional voidratio determination.
Physics of Fluids , , 055101.Barbaca, L., Pearce, B. W., Ganesh, H., Ceccio, S. L., & Brandner, P. A.(2019). On the unsteady behaviour of cavity flow over a two-dimensionalwall-mounted fence. Journal of Fluid Mechanics , , 483–525.Belahadji, B., Franc, J.-P., & Michel, J.-M. (1995). Cavitation in the rotationalstructures of a turbulent wake. Journal of Fluid Mechanics , , 383–403.Bernal, L., & Roshko, A. (1986). Streamwise vortex structure in plane mixinglayers. Journal of Fluid Mechanics , , 499–525.30iswas, G., Breuer, M., & Durst, F. (2004). Backward-facing step flows forvarious expansion ratios at low and moderate reynolds numbers. J. FluidsEng. , , 362–374.Bradshaw, P., & Wong, F. (1972). The reattachment and relaxation of a tur-bulent shear layer. Journal of Fluid Mechanics , , 113–135.Brennen, C. E., & Brennen, C. E. (2005). Fundamentals of multiphase flow .Cambridge university press.Callenaere, M., Franc, J.-P., Michel, J.-M., & Riondet, M. (2001). The cavi-tation instability induced by the development of a re-entrant jet.
Journal ofFluid Mechanics , , 223–256.Driver, D. M., Seegmiller, H. L., & Marvin, J. G. (1987). Time-dependentbehavior of a reattaching shear layer. AIAA journal , , 914–919.Eaton, J., & Johnston, J. (1981). A review of research on subsonic turbulentflow reattachment. AIAA journal , , 1093–1100.Franc, J.-P., & Michel, J.-M. (1983). Two-dimensional rotational structures inthe cavitating turbulent wake of a wedge.Ganesh, H., M¨akiharju, S. A., & Ceccio, S. L. (2016). Bubbly shock propagationas a mechanism for sheet-to-cloud transition of partial cavities. Journal ofFluid Mechanics , , 37–78.Hasan, M. (1992). The flow over a backward-facing step under controlled per-turbation: laminar separation. Journal of Fluid Mechanics , , 73–96.Iyer, C. O., & Ceccio, S. L. (2002). The influence of developed cavitation onthe flow of a turbulent shear layer. Physics of fluids , , 3414–3431.Jovic, S., & Driver, D. (1995). Reynolds number effect on the skin friction inseparated flows behind a backward-facing step. Experiments in Fluids , ,464–467. 31atz, J., & O’hern, T. (1986). Cavitation in large scale shear flows, .O’hern, T. (1990). An experimental investigation of turbulent shear flow cavi-tation. Journal of fluid mechanics , , 365–391.Scarano, F., Benocci, C., & Riethmuller, M. (1999). Pattern recognition analysisof the turbulent flow past a backward facing step. Physics of Fluids , , 3808–3818.Spazzini, P., Iuso, G., Onorato, M., Zurlo, N., & Di Cicca, G. (2001). Unsteadybehavior of back-facing step flow. Experiments in fluids , , 551–561.Troutt, T., Scheelke, B., & Norman, T. (1984). Organized structures in areattaching separated flow field. Journal of Fluid Mechanics , , 413–427.Wee, D., Yi, T., Annaswamy, A., & Ghoniem, A. F. (2004). Self-sustainedoscillations and vortex shedding in backward-facing step flows: Simulationand linear instability analysis. Physics of Fluids , , 3361–3373.Wu, J. (2019). Bubbly Shocks in Separated Cavitating Flows . Ph.D. thesis.Wu, J., Ganesh, H., & Ceccio, S. (2019). Multimodal partial cavity shedding ona two-dimensional hydrofoil and its relation to the presence of bubbly shocks.
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