Wall modeling via function enrichment: extension to detached-eddy simulation
WWall modeling via function enrichment: extension to detached-eddy simulation
Benjamin Krank, Martin Kronbichler, Wolfgang A. Wall ∗ Institute for Computational Mechanics, Technical University of Munich,Boltzmannstr. 15, 85748 Garching, Germany
Abstract
We extend the approach of wall modeling via function enrichment to detached-eddy simulation. The wall model aims at usingcoarse cells in the near-wall region by modeling the velocity profile in the viscous sublayer and log-layer. However, unlike otherwall models, the full Navier–Stokes equations are still discretely fulfilled, including the pressure gradient and convective term. Thisis achieved by enriching the elements of the high-order discontinuous Galerkin method with the law-of-the-wall. As a result, theGalerkin method can “choose” the optimal solution among the polynomial and enrichment shape functions. The detached-eddysimulation methodology provides a suitable turbulence model for the coarse near-wall cells. The approach is applied to wall-modeled LES of turbulent channel flow in a wide range of Reynolds numbers. Flow over periodic hills shows the superioritycompared to an equilibrium wall model under separated flow conditions.
Keywords:
Delayed detached-eddy simulation, Spalart–Allmaras model, Function enrichment, High-order discontinuous Galerkin
1. Introduction
Wall modeling via function enrichment is a spatial discretiza-tion technique that allows the resolution of the sharp bound-ary layer gradients present in high-Reynolds-number flows withrelatively coarse meshes. The basic idea is to make use of theflexibility in Galerkin methods regarding the choice of the solu-tion space: a few additional, problem-tailored shape functionsare used to approximate the solution, in addition to the commonpolynomials. Using these enriched elements, the full Navier–Stokes equations are solved in the whole boundary layer in aconsistent manner. As a result, the wall model can take intoaccount high adverse pressure gradients and convective e ff ects,unlike most other wall modeling approaches.The idea of wall modeling via function enrichment was pro-posed by Krank and Wall [1] within the continuous Galerkinmethod (standard FEM) as a wall modeling technique for large-eddy simulation (LES). While that work showed promising re-sults in separated flows, the limiting factor in terms of accuracywas the turbulence model employed in the near-wall region.A residual-based approach was used, supported by a structuralLES model in the outer layer, a model that was originally notintended for underresolved boundary-layer simulations. Thewall modeling approach was since applied in conjunction withRANS [2] employing the Spalart–Allmaras (SA) model withinthe high-order discontinuous Galerkin (DG) method. ∗ Corresponding author at: Institute for Computational Mechanics, Techni-cal University of Munich, Boltzmannstr. 15, 85748 Garching, Germany. Tel.: +
49 89 28915300; fax: +
49 89 28915301
Email addresses: [email protected] (Benjamin Krank), [email protected] (Martin Kronbichler), [email protected] (Wolfgang A. Wall)
In this article, we show that the widely used delayed detached-eddy simulation (DDES) methodology [3] may be used to modelthe unresolved turbulence in the near-wall region in wall mod-eling via function enrichment. This can be done by extendingthe implementation of the SA model [2] in a straightforwardway. The idea of the original DES approach [4] is that the walldistance function y present in the SA model is limited with acharacteristic cell length ∆ according to y DES = min( y , C DES ∆ ) , (1)where the parameter C DES has been calibrated to C DES = . ∆ = max( ∆ x , ∆ y , ∆ z ) [5]. Asa result, the RANS model acts as a one-equation LES subgridmodel if y > C DES ∆ . DDES represents an enhancement of thatmethodology by defining the wall-distance parameter as y DDES = y − f d max(0 , y − C DES ∆ ) , (2)with the functions f d = − tanh (cid:16) (8 r d ) (cid:17) , (3) r d = ν + ν t (cid:112) ( ∇ u ) i j ( ∇ u ) i j κ y , (4)where u is the velocity vector, ν the kinematic and ν t the eddyviscosity, and κ = .
41. (D)DES is widely used in researchand industry, see, e.g., [6, 7] and is today even used for theaerodynamics of entire vehicles [8] due to its good accuracyin separated flows and the ability to investigate acoustic noisesources in the flow. Regarding the application of DES, twomain branches are frequently used. The original idea was tosimulate the whole boundary layer in RANS mode and to com-pute free shear layers in LES mode only [4]. As an alternative,
Preprint submitted to Computers & Fluids December 25, 2017 a r X i v : . [ phy s i c s . f l u - dyn ] D ec ES can be seen as an approach to wall-modeled LES (WM-LES), in which only the inner layer is computed in RANS modeand the outer boundary layer in LES mode, see, e.g., [9].Wall modeling via function enrichment has the potential ofsignificantly reducing the computational cost of (D)DES. Thegrid saving of the standard (D)DES in comparison to LES isachieved by using relatively coarse meshes in the wall-paralleldirections of up to 0 . δ (WMLES) and δ (classical DES) [10]with the boundary layer thickness δ . The wall-normal direc-tion necessitates many grid points in order to resolve the lami-nar sublayer due to the requirement of placing the first o ff -wallnode at y + ∼
1, however. For example, if a boundary layerof a thickness of 10 ,
000 wall units is computed with a con-stant grid stretching factor of 1.15 [9], a total of 53 grid layerswould be required. This is a quite high cost compared to therelatively low engineering interest in that region. Wall mod-eling via function enrichment allows the first grid point to belocated in the range y + ∼
10 to 100, saving 17–33 grid layersfor that example, without noteworthy loss in accuracy, in addi-tion to much better conditioned equation systems through thelower grid anisotropy.In the next section, we give details on how the enrichmentshape functions are constructed. In Section 3, the high-orderDG code employed for the validation is outlined and numericalexamples are presented in Section 4.
2. Wall modeling via function enrichment
The primary idea of wall modeling via function enrichmentis as follows. In a single element row at the wall, the discrete ve-locity solution u h is composed of two parts, the standard poly-nomial component, ¯ u h , and an additional enrichment compo-nent, (cid:101) u h , yielding u h ( x , t ) = ¯ u h ( x , t ) + (cid:101) u h ( x , t ) . (5)The polynomial component is given in each cell as an FE-expansionaccording to ¯ u h ( x , t ) = (cid:88) B ∈ N k N kB ( x ) ¯ u B ( t ) (6)with the shape functions N kB of polynomial degree k and cor-responding degrees of freedom ¯ u B . There are several ways ofconstructing the enrichment component. In its simplest form,an enrichment function ψ is weighted in each element with oneadditional node (cid:101) u , i.e., one degree of freedom per space di-mension, with (cid:101) u h ( x , t ) = ψ ( x , t ) (cid:101) u ( t ) . (7)The enrichment function can additionally be weighted using alow-order polynomial to yield a higher level of flexibility inthe function space [1, 2], which is not considered herein. It isthis enrichment function that is responsible for the e ffi ciencyof the approach. By taking ψ as a wall function, the solutionspace of the Galerkin method is capable of resolving a sharpattached boundary layer with very few degrees of freedom. It isnoted that this wall function is not prescribed as a solution, butthe Galerkin method automatically “chooses” the best possible solution within the high-order polynomials and the enrichmentcomponent in a least squares sense. As a wall function, weconsider Spalding’s law [11] in the form y + = ψκ + e − κ B (cid:32) e ψ − − ψ − ψ − ψ − ψ (cid:33) , (8)with κ = .
41 and B = .
17, as it was implemented in [1].Several alternative wall functions have been discussed in [2]. Inthe wall-normal direction, Spalding’s law scales with the wallcoordinate y + = yu τ /ν with u τ = (cid:112) τ w /ρ and the density ρ suchthat the wall shear stress τ w is represented correctly. In turn,the wall function has to be adapted according to the local wallshear stress in the numerical method, and its temporal evolutionhas to be taken into account.We have developed an algorithm in [1, 2], which enablessuch an adaptation. Therein, the wall shear stress is computedon discrete nodes via the velocity derivative according to τ w , B = (cid:107) (cid:82) ∂ Ω D N c , mB ( x ) ρν ∂ u h ∂ y (cid:12)(cid:12)(cid:12) y = d A (cid:107) (cid:82) ∂ Ω D N c , mB ( x ) d A , (9)with linear continuous shape functions N c , mB of degree m = τ w , h = (cid:88) B ∈ N c , m N c , mB τ w , B , (10)yielding a continuous representation of the wall shear stress.Through the choice of m =
1, the wall shear stress τ w , h is acoarsened field, since usually k >
1; the coarsening is manda-tory because the wall functions are relations for the mean quan-tities, meaning that the mean wall shear stress is related to themean velocity, and the average wall shear stress would other-wise be overpredicted, see Reference [1]. This field is updatedprior to each time step, such that the function space of the ve-locity changes continuously and adapts to the local flow con-ditions. Further details on the adaptation algorithm are givenin [2]. Near separation or reattachment locations, it may hap-pen that the wall shear stress becomes zero, which renders thefunction space linear dependent. However, considering that thefirst o ff -wall point is located very close to the wall in terms of y + at these locations, a simple and consistent solution is to tempo-rally “switch o ff ” the enrichment in the respective cells, see [2]for details and [12] for an evaluation of the method in WMLESin the context of another turbulence modeling approach. If wallshear stress becomes larger at these locations at a later instance,the enrichment is “switched on” again.Finally, we comment on the two additional variables, whichhave to be discretized: the pressure and the working variable ofthe SA model. Both variables do not exhibit high gradients atthe wall, such that they are represented su ffi ciently well by thestandard FE space only, according to [2].
3. Numerical method
The present wall modeling approach may be implementedin any FEM and DG flow solver. In this work, we consider the2mplementation of the incompressible Navier–Stokes equationswith the SA model in [2], which in turn is based on the incom-pressible high-performance high-order semi-explicit DG codeINDEXA [13]. An extension of the present wall modeling ap-proach to the compressible Navier–Stokes equations would bestraightforward, since high gradients are commonly not presentin the energy variable, such that the latter may be consideredanalogous to the pressure variable herein. Numerical methodsbased on the continuous FEM would require a small modifica-tion of the enrichment component as described in [1].The solver is based on weak forms, which are described indetail in [2]. These weak forms include volume and surfaceterms, that have to be integrated over cells and faces. The in-tegrals are in our solver evaluated using the high-performancekernels by Kronbichler and Kormann [14] within the deal.II fi-nite element library [15]. In particular, the integrals have poly-nomilal and nonpolynomial paths, the latter due to the nonpoly-nomial character of the enrichment function. The polynomialpaths are integrated using the quadrature formulas given in [13]and are evaluated exactly on a ffi ne cells. The nonpolynomialcontributions have to be evaluated with more quadrature points,in particular in the wall-normal direction [1]. From our exten-sive experience with wall modeling via function enrichment,we can give the following guide lines: If the enriched cells ex-tend up to approximately y + e =
90 in the statistical quantities,8 quadrature points in the wall-normal direction are su ffi cient.Further we have y + e <
110 (10 points), y + e =
130 (12 points), y + e =
200 (17 points); see also the monograph [16] for furtherdetails. All simulation cases presented herein use an adaptivetime stepping method presented in [2] with a temporal accuracyof second order, a Courant number of Cr = .
14, and a di ff usionnumber of D = .
02. In the particular formulation used with theenrichment, the solver has a formal spatial order of accuracy of k . Finally, we note that we apply no-slip boundary conditionsweakly according to [13] in all steps of the scheme for the ex-amples presented in this article, which limits the width of thefirst o ff -wall cell to a few hundred wall units, as the no-slipcondition would otherwise be violated severely.The increasing resolution power of the DG scheme with in-creasing polynomial degree should be taken into account in the(D)DES grid length scale ∆ [17]. Based on the analysis of theresolution power of DG schemes performed in [18], we choose ∆ = ∆ e k + ∆ e , in contrastto the choice of the factor of 1 / k chosen in Reference [17].
4. Numerical examples
Wall modeling via function enrichment is assessed by con-sidering DDES in the WMLES branch. In the first example, weinvestigate the method for attached equilibrium boundary layerflows present in turbulent channel flow. The second exampleconsiders flow over periodic hills in order to analyze the behav-ior of the enrichment in conjunction with DDES in a nonequi-librium flow. As a result of earlier studies [13, 19], the poly-
Table 1: Overview of simulation cases for the turbulent channel flow. Thenumber of polynomial grid points per direction i is N i = ( k + N ie with thenumber of cells per direction N ie and the polynomial degree k = ∆ y + e is thethickness of the first o ff -wall cell, in which the enrichment is active, y = C DES ∆ is the RANS–LES switching location in terms of channel half-height δ , anderr( τ w ) is the relative error of the computed wall shear stress. Re τ N e × N e × N e γ ∆ y + e y = C DES ∆ err( τ w )395 16 × × . . δ . × × . . δ . ,
000 16 × × . . δ − . × × . . δ . × ×
16 1 . . δ . ,
200 16 × × . . δ . ,
000 16 × × . . δ − . ,
000 16 × × . . δ . ,
000 16 × × . . δ − . Figure 1: Mesh for turbulent channel flow at Re τ = th degreeplus one enrichment shape function in the enriched cells.Figure 2: Instantaneous numerical solution of turbulent channel flow at Re τ =
950 via velocity magnitude. Red indicates high and blue low values. nomial degree of k = We consider flow in a stream- and spanwise periodic chan-nel of the dimensions 2 πδ × δ × πδ in streamwise, wall-normal,and spanwise direction, respectively, with the channel half-height δ . The flow is driven by a constant body force, which is derivedfrom the nominal quantities. We investigate this flow in a widerange of friction Reynolds numbers Re τ = u τ δ/ν , which arechosen according to the available DNS data at Re τ =
395 [20], Re τ =
950 [21], Re τ = ,
000 [22], Re τ = ,
200 [23], and ad-ditionally Re τ = , Re τ = , Re τ = , y + u + Re τ = 10 , Re τ = 20 , Re τ = 50 , Re τ = 395 Re τ = 950 Re τ = 2 , Re τ = 5 , Re τ = 2 , u h full solution e u h enrichment solutionDNSlin/log-law × × × × × × y + u ′ + Re τ = 395 Re τ = 950 Re τ = 2 , Re τ = 5 , y + v ′ + y + w ′ + x / δ + 1 ( u ′ v ′ ) + Figure 3: DDES (WMLES) of turbulent channel flow at several Reynolds numbers. Mean velocity (left) and RMS-velocities as well as Reynolds shear stress (right).All quantities are normalized according to u + = (cid:104) u (cid:105) / u τ , u (cid:48) + = (cid:113) (cid:104) u (cid:48) (cid:105) / u τ , v (cid:48) + = (cid:113) (cid:104) u (cid:105) / u τ , w (cid:48) + = (cid:113) (cid:104) u (cid:105) / u τ , and ( u (cid:48) v (cid:48) ) + = (cid:104) u u (cid:105) / u τ . grid length scale yields approximately ∆ = . δ for mostcases, so the RANS–LES switching point is located at C DES ∆ = . δ . One simulation case uses twice the number of grid cellsin streamwise and spanwise direction, resulting in a RANS–LES switching point near C DES ∆ = . δ . As for the wall-normal resolution, the enrichment is taken into account in thewall-nearest cell layer in all simulation cases, see Figure 1. Asit was discussed earlier, the enrichment shape functions allowthe resolution of the averaged near-wall flow with very coarsecell sizes. The width of the first o ff -wall cell lies in this work inthe range of 51 to 191 wall units. In order to enable an applica-tion to high Reynolds numbers, a hyperbolic grid stretching isadditionally considered, according to f : [0 , → [ − δ, δ ]: x (cid:55)→ f ( x ) = δ tanh( γ (2 x − γ ) , (12)with the mesh stretching parameter γ . The values of γ forall simulation cases are included in Table 1. In the numericalmethod, the velocity solution is postprocessed at a large num-ber of wall-normal layers inside each cell using the definitionof the velocity variable (5) such that the behavior of the enrich- ment may be analyzed. Statistics were acquired in a simulationtime interval of approximately 60–95 flow-through times basedon a fixed time interval.The turbulent flow is visualized at one time instant in Fig-ure 2. Time-averaged results are presented in Figure 3. Therein,the results are plotted in terms of the normalized mean velocity u + = (cid:104) u (cid:105) / u τ , the RMS velocity components u (cid:48) + = (cid:113) (cid:104) u (cid:48) (cid:105) / u τ , v (cid:48) + = (cid:113) (cid:104) u (cid:105) / u τ , and w (cid:48) + = (cid:113) (cid:104) u (cid:105) / u τ , as well as the Reynoldsshear stress ( u (cid:48) v (cid:48) ) + = (cid:104) u u (cid:105) / u τ , which are all normalized us-ing the numerical value of u τ . The mean velocity is generallypredicted very accurately in the laminar sublayer and the log-layer, where the enrichment shape functions are active. In or-der to get a better impression of the role of the enrichment, thenumerical enrichment solution is plotted in Figure 3 alongsidethe full mean velocity solution. The enrichment solution repre-sents the largest part of the near-wall solution in most cases,including the high velocity gradient. In particular in cases,where the first o ff -wall cell spans a range of more than 100wall units, the enrichment is the main contributor to the meanvelocity. Solely at the lowest Reynolds number, the enrichment4 able 2: Simulation cases and resolutions of the periodic hill flow. The cases use a coarse mesh with 32 × ×
16 grid cells and a fine mesh with 64 × ×
32 elements.The polynomial degree is k = k + x , sep and x , reatt correspond to the zero-crossings of the skin friction. Case N e × N e × N e N × N × N Re H max( ∆ y + e ) x , sep / H x , reatt / H ph10595 coarse 32 × ×
16 160 × ×
80 10 ,
595 76 0 .
25 4 . × ×
32 320 × ×
160 10 ,
595 36 0 .
16 4 . × ×
448 10 ,
595 - 0 .
19 4 . × ×
16 160 × ×
80 37 ,
000 144 0 .
40 3 . × ×
32 320 × ×
160 37 ,
000 79 0 .
26 4 . ,
000 - - 3 . × ×
64 37 ,
000 - - 2 . × ×
128 37 ,
000 - - 2 . x x Figure 4: Mesh for flow over periodic hills of the case ph37000 coarse. Redindicates enriched cells and blue standard polynomial cells, i.e., a single layerof cells at the wall is enriched. In each cell, the solution consists of a polynomialof 4 th degree plus one enrichment shape function in the enriched cells. x x Figure 5: Instantaneous numerical solution of flow over periodic hills of thecase ph37000 coarse via velocity magnitude. Red indicates high and blue lowvalues. solution plays a minor role, which essentially means that thepolynomial component is capable of resolving most of the flow.Further away from the wall we observe the characteristic log-layer mismatch, that we expect in wall-attached simulations us-ing DDES [9, 27]. The log-layer mismatch is especially visiblefor the lower Reynolds numbers. We note that there are sev-eral techniques available in the literature that reduce this e ff ect,for example [28]. In the framework of the present enrichmentmethodology, it is possible to construct an alternative hybridRANS / LES turbulence model, which does not show a log-layermismatch by definition. We have recently developed such anapproach, which is the topic of a subsequent publication [12].The RMS velocities and the Reynolds shear stress are alsopresented in Figure 3 up to Re τ = ,
200 and compared withthe DNS data. These quantities show that the RANS–LES tran-sition extends up to approximately 0 . δ and the flow is in fullLES mode further away from the wall. This means that we do ∆ y + e x /H ph10595 coarseph10595 fi ne ∆ y + e x /H ph37000 coarseph37000 fi ne Figure 6: Width of wall-layer (width of first o ff -wall cell) for Re H = , Re H = ,
000 (bottom). The shallower curves correspond to theupper wall. not expect agreement with the DNS below 0 . δ , and the curvesmatch the DNS above this value very well. Only in the refinedcase at Re τ = , ffi cient presentedin [9] of up to 22%, this is an excellent result.We conclude from this section that wall modeling via func-tion enrichment allows an accurate computation of the near-wall region in turbulent boundary layers with very coarse cells,while still computing the full incompressible Navier–Stokes equa-5 x /H c f x /H c p ph10595 coarseph10595 fi neKKW DNS Figure 7: Skin friction coe ffi cient at the lower wall (left) and pressure coe ffi cient at the lower and upper boundary (right). The shallower pressure coe ffi cient curvescorrespond to the upper wall. tions in the whole boundary layer. DDES is a suitable turbu-lence modeling approach for wall modeling via function en-richment. As a second benchmark example, we consider flow over pe-riodic hills at the Reynolds numbers based on the hill height H and bulk velocity u b of Re H = ,
595 and Re H = , / LES methods were assessed using this flowconfiguration within the European initiative “Advanced Tur-bulence Simulation for Aerodynamic Application Challenges”(ATAAC) [29], including DDES (see the final report by Jakirli´cfor cross-comparison of results). A strong adverse pressure gra-dient and flow separation from the curved boundary are chal-lenging for many statistical modeling approaches, but DDESyielded very good agreement with a reference LES in that study.Also, all previous publications on wall modeling via functionenrichment [1, 2, 12] used this benchmark example, and verypromising results were obtained if a turbulence resolving ap-proach was used. Reference data for this flow is provided byDNS at the lower Reynolds number [19] (available for down-load at [24]) and water-channel experiments [25] at the higherReynolds number.The computational domain is of the dimensions 9 H × . H × . H in streamwise, vertical and spanwise direction, respectively, andthe lower wall is given by the smoothly curved hill shape. Thedomain is extended periodically in the streamwise and spanwisedirection, and no-slip boundary conditions are applied on theupper and lower wall. The computational setup is very similarto the simulations of the DNS [19]. Two meshes are consid-ered at each Reynolds number, a coarser mesh with 32 × × × ×
32 cells. As for the previousexample, the solution is represented by a polynomial of degree4 in each cell, plus one enrichment shape function in the wall-nearest cell layer. The mesh is moderately stretched towardsthe no slip walls to yield a better resolution of the near-wallarea, and the geometry is mapped onto the exact hill shape us-ing an isogeometric approach. One representative mesh is dis-played in Figure 4. The wall-normal width of the enrichmentlayer is plotted in Figure 6 in wall coordinates. An overview of all simulation cases and resolution parameters is given in Ta-ble 2. Statistics were averaged in a simulation time interval of61 flow-through times. One snapshot of the instantaneous ve-locity field is visualized in Figure 5.We begin the discussion of the results with the skin frictionand pressure coe ffi cients c f and c p . They are defined as c f = τ w ρ u b , c p = p − p ref12 ρ u b , where the reference pressure p ref is taken at x = ffi cient predicted bythe coarse mesh shows an overprediction of the magnitude be-tween x / H = x / H =
4. Even the characteristic peakin the skin friction on the windward side of the hill crest is pre-dicted very well for both cases. The overall excellent agreementis also observed in the estimation of the length of the reattach-ment zone of x , reatt / H = .
51 and 4 .
40 (see Table 2) in com-parison to the DNS result of x , reatt / H = . ± . ff erence at x / H = Re H = , u/u b + x /H x / H x / H v/u b + x /H u ′ v ′ /u b + x /H x / H K/u b + x /H x / H ph10595 coarseph10595 fi ne KKW DNS
Figure 8: Streamwise u = (cid:104) u (cid:105) and vertical v = (cid:104) u (cid:105) mean velocity, Reynolds shear stress u (cid:48) v (cid:48) = (cid:104) u u (cid:105) − (cid:104) u (cid:105)(cid:104) u (cid:105) , and turbulence kinetic energy K = / u (cid:48) u (cid:48) + v (cid:48) v (cid:48) + w (cid:48) w (cid:48) ) of the periodic hill flow at Re H = , mentation of an equilibrium wall model within the high-orderDG [26] (cases baseline and fine in that publication). Thesesimulations employ grids comparable to the respective coarseand fine case presented in this work and are also included inthe overview if Table 2. Regarding the mean velocity, all wall-modeled cases yield larger errors as compared to the lower Reynoldsnumber. The equilibrium wall model overpredicts the veloc-ity in the recirculation zone, yielding a shorter reattachment length of x , reatt / H = . x , reatt / H = . x , reatt / H = .
76, see Table 2). The presentwall-enriched DDES simulations overpredict the mean stream-wise velocity in that region with the coarse mesh and under-predict the velocity in the fine case. Yet, the DDES cases arecloser to the reference than the equilibrium model, both for thecoarse and fine mesh. The reattachment lengths are computedas x , reatt / H = .
37 and 4 .
53 and confirm the observations of the7 u/u b + x /H x / H x / H v/u b + x /H u ′ v ′ /u b + x /H x / H K/u b + x /H x / H ph37000 coarseph37000 fi ne RM Exp
CM WMLES coarseCM WMLES fi ne Figure 9: Streamwise u = (cid:104) u (cid:105) and vertical v = (cid:104) u (cid:105) mean velocity as well as Reynolds shear stress u (cid:48) v (cid:48) = (cid:104) u u (cid:105)−(cid:104) u (cid:105)(cid:104) u (cid:105) of the periodic hill flow at Re H = , mean velocity. The profiles of the vertical velocity yield di ff er-ences with the reference on the lee side of the hill as a resultof the di ff erent length of the separation bubble. The magnitudeof the Reynolds shear stress is overpredicted by the coarse caseand is accurately estimated by the fine case.We conclude from the results of the periodic hill flow thatwall modeling via function enrichment with DDES as turbu-lence model is well capable of computing nonequilibrium flows.This is due to the full consistency of the method, as all terms ofthe Navier–Stokes equations are satisfied discretely.
5. Conclusions
In this work, we have used the DDES methodology to modelthe unresolved turbulent motions in wall modeling via functionenrichment. The idea of this wall model is that an additionalshape function is included in each cell, which has the shape of awall function. As a result, the Galerkin method can resolve typ-ical attached boundary layer profiles with very coarse meshes. Since the standard high-order polynomial shape functions arestill available in all cells, the method is su ffi ciently flexible torepresent nonequilibrium boundary layers with a high pressuregradient and separated boundary layers.Wall modeling via function enrichment with the DDES tur-bulence model does not provide a solution to the problems inthe hybrid RANS–LES transition region in attached boundarylayers. However, an alternative hybrid RANS / LES turbulencemodeling approach can be constructed based on the enrichment,which a priori circumvents these problems and the associatedlog-layer mismatch. This turbulence model is described in afollow-up paper [12].
Acknowledgements
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