Evaluation of the unsteady RANS capabilities for separated flows control

E. Garnier, P.Y. Pamart, J. Dandois, P. Sagaut

Evaluation of the unsteady RANS capabilities for separated flows control

Computers & Fluids , Vol. 61, 2012

108

Evaluation of the unsteady RANS capabilities for separated flows control

E. Garnier^a,⇑, P.Y. Pamart^a, J. Dandois^a, P. Sagaut^b

aONERA, Applied Aerodynamics Department, 8 rue des Vertugadins, 92190 Meudon, France

bIJLRA/UPMC, 4 Place Jussieu, 75252 Paris Cedex 5, France

a r t i c l e i n f o

Article history:

Received 3 February 2011

Received in revised form 18 August 2011 Accepted 24 August 2011

Nowadays, unsteady RANS methods are the only techniques usable in the industry to design a control device based on a periodic excitation of the flow. This paper aims at evaluating the potential of such methods by comparing their performances with respect to LES computations taken as a reference. The chosen test case is a rounded backward facing step proposed by Dandois et al.[1]. The frequency response of this flow to the periodic excitation of a synthetic jet is computed both with LES and with four different URANS models. It is demonstrated that the optimal frequency identified by best URANS models is close to the one of LES. Nevertheless, the amplitude of separated bubble surface reduction is underestimated by URANS models.

Ó2011 Elsevier Ltd. All rights reserved.

1. Introduction

Flow control is subjected to a growing interest and some indus-trials begin to seriously consider implementing flow control de-vices such as pulsed or synthetic jets to manage flow separation.

Possible benefit[2]may be for example the suppression of slats or of the gap between wing main body and flaps. These studies should rely on accurate prediction tools and, up to now, the only methods employed routinely in the industry to lead a design pro-cess are based on RANS/URANS techniques. For the cases of pulsed and synthetic jets which generate periodic excitation, the only usable technique in an industrial perspective is URANS. The goal of this paper is to assess URANS capabilities for the control of a sep-arated configuration. In particular, one of the main issue is to as-sess the capacity of URANS methods to reproducing with enough accuracy the frequency response of the flow forced by a periodic excitation. This study is a sequel of Refs.[1,3]in which large eddy simulation (LES) was used to compute the separated flow over a rounded backward facing step controlled by a synthetic slot. This configuration is interesting in many aspects. First, LES has been validated against DNS [1] so reference solutions can now be computed with LES in order to compare with URANS results. Sec-ondly, the frequency effect has already been assessed with LES and a comprehensive database is available with 13 computations in the range of non dimensionalized frequency varying from 0.1 to 10[3]. This database is available for comparison with URANS techniques. Thirdly, the test case is both numerically simple enough (bi-dimensional, with well defined boundary conditions)

and representative enough to assess a large number turbulence models while drawing relevant conclusions. In terms of turbulence modeling, it is worthwhile to notice that the focus is mainly set on the assessment of models widely used in the industry. Cross com-parisons between RANS and LES have already been performed in the past. For example, Rumsey has summarized in Ref.[4]the re-sults of the CFDVAL workshop. For a case where a synthetic jet interacts with a canonical boundary layer, it was shown that UR-ANS is able to reproduce correctly the mean field even if turbulent structures present in the shear layer are filtered. This conclusion is partly based on the study presented in Ref.[5]but does not con-cern separated flows. A separated flow test case (case 3 of the CFD-VAL workshop) which resembles to the one considered here has already been treated with LES (see for example Ref.[6]) and URANS but to our knowledge, the frequency effect has never been investi-gated on this test case.

The paper is organized as follows. The test case is first de-scribed. Then, a brief description of the numerical methods is pro-vided. The results section discusses of the frequency response of the flow essentially on the mean field. In particular, the surface of the separation bubble is chosen as a measure of the URANS computations fidelity.

2. Test case description 2.1. Description the configuration

The topology of the configuration is displayed in Fig. 1. The shape of the rounded backward-facing step is the same as the one defined in Dandois et al. [1]. The step heighth is equal to 20 mm and the maximum slope of the step is equal to 35 degrees.

0045-7930/$ - see front matterÓ2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2011.08.016

⇑ Corresponding author.

E-mail address:eric.garnier@onera.fr(E. Garnier).

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The inflow static pressureP₁and static temperatureT₁are equal to 19,501 Pa and 281 K respectively. The Mach numberM1is set equal to 0.33 and the free-stream velocity U1 is equal to 109.6 m s¹. The boundary layer thickness d at x/h=1 and its momentum thicknesshare equal to 0.5hand 0.05hrespectively.

The Reynolds number based on the momentum thicknessRehtakes a value equal to 1350 at the separation point. The Reynolds num-ber based on the step height and free-stream velocityRehis equal to 30,000. The synthetic jet orifice consists in a two-dimensional slot. Its windward edge is located near to the mean separation point of the uncontrolled flow as in the paper of Neumann and Wengle[7]. The coordinate system is the following:xis oriented in the streamwise direction,y denotes the vertical direction and z designates the spanwise direction. The origin is located at the beginning of the ramp.

2.2. Actuator description

The actuator is composed of a slot cavity (displayed inFig. 1) and is described by four parameters: slot widthd, slot heightHs, cavity width Wc and cavity height Hc. The actuator dimensions used in the present computations ared= 0.55 mm,Hs=d,Wc= 2d andHc=d/2.

Actuator dimensions have been calculated on the one hand to satisfy the jet formation criteria proposed by Holman et al. [8]

and on the other hand to set the resonance frequency out of the frequency interval of interest (at value ofF⁺59). The synthetic jet forcing amplitude is characterized by the momentum coeffi-cient defined in Eq.(1).

Cl¼2qjdV²_rms

q₁LncU²₁ ð1Þ

whereVrmsis the root-mean-square value of the synthetic jet ampli-tude velocity at the orifice exit andLncis the separation length of the uncontrolled case for the LES computation.

To simulate the diaphragm displacement, a blowing/suction condition with a top-hat spatial distribution and sinusoidal temporal variation is implemented on the whole cavity bottom surface:V(x,t) =Vactcos(2pf t). This boundary condition has been successfully validated in a previous study[5]. In the present study, Vacthas been set equal to 19.9 m/s. This gives aClvalue equal to 0.33%.

The reduced frequency work range is chosen within the interval F⁺2[0.1; 10] with:F⁺=f Lnc/U1. Quantitatively,F⁺equal to 1 corre-sponds to a frequencyfequal to 1242 Hz. SinceLncis model-depen-dent, it has been preferred to carry out controlled simulation at the same dimensional frequency. Thirteen frequencies has been assessed:

f2 f123;369;615;861;1230;1500;1740;1845;2460;

3690;4920;7380; 12;320gHz:

This frequency range is large enough to cover theF⁺O(1) con-trol preferred in Refs.[9,10]and theF⁺O(10) promoted for exam-ple by Glezer et al.[11].

2.3. Numerical method

Two codes have been used in this study. The FLU3M code is the solver with which most of LES computations have been done at ONERA in the last decade. Nevertheless, the only RANS model available in this tool is the Spalart–Allmaras model. Conversely, the more recent elsA solver[12]makes available a large variety of turbulence models. For these reasons, FLU3M has been used for LES computations and elsA has been preferred for URANS com-putations. Both codes are finite-volume solvers for the compress-ible Navier–Stokes equations. In both cases, the time integration is carried out by means of the second order-accurate backward scheme of Gear. More details about this algorithm are available in Péchier et al.[13]. The time step is taken equal to 0.00274 h/

U₁both for RANS and LES simulations. For the highest frequency, it represents 162 time steps per period. It is sufficient to avoid any significant results dependence on the time step. For LES simula-tions, the spatial scheme is the one proposed by Mary and Sagaut [14]. For RANS computations which tolerate a higher level of numerical dissipation, the Jameson scheme has been employed.

For this scheme, the second and fourth order dissipation coeffi-cients have been set respectively equal to 0.5 and 0.016. The grid resolution is so fine for a URANS study that a sensitivity to the spatial scheme choice should not be expected. For both codes, the viscous fluxes are computed with a second order accurate cen-tered scheme. It has been checked that, with the Spalart–Allmaras model, FLU3M and elsA give the same results despite a different spatial scheme.

Fig. 1.Grid used for LES computations.

For the LES computations, the grid is composed of a three-dimensional grid which encompasses the turbulent region and a two-dimensional one in which the flow is quasi-potential in order to restrict the number of computational points. The streamwise length of the computational domain is 24 h (7.5 h upstream of the separation point and 12 h downstream from the reattachment point of the uncontrolled case), its spanwise extent is 4 h and its height is 10 h in the inflow plane. The grid is composed of 1210⁶cells. Grid spacings are equal toDx^þ¼50; Dy^þ_min¼0:5 and Dz⁺= 18 in the inflow boundary layer. In the separated zone, grid spacing in the streamwise direction is always smaller than 50 wall units.

A view of the LES grid in thex–yplane is displayed inFig. 1. The modified synthetic eddy method for generating turbulent inflow boundary conditions proposed by Pamiès et al.[15]has been em-ployed for LES computations. It allows the introduction of realistic

putational cost compared to a simulation in which the laminar/tur-bulent transition is computed.

For the URANS simulations, the domain has been lengthened in the streamwise direction so that the boundary layer has en-ough distance to reach the same thickness as the one of the LES at the beginning of the ramp. The necessary development distance of the boundary layer is equal to 36 h. Using this strat-egy, a uniform inflow condition is sufficient for URANS simula-tions. Apart from this modification, the grid is the same as the LES one downstream from x=7 h. Of course, only two longitu-dinal planes are necessary for RANS computations and the num-ber of points is limited to 0.13410⁶. This makes URANS computations much cheaper than LES ones. Besides, it is worth noticing that the mean profile set at the entrance of the compu-tational domain for LES simulations is the same as for RANS sim-ulations at the location x/h=7. A plot of the URANS grid is available inFig. 2.

Fig. 2.Grid used for RANS computations.

Fig. 3.Strioscopy of the flow in the symmetry plane for the uncontrolled LES case (black & white). The mean flow is superimposed (colors). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4.Strioscopy of the flow in the symmetry plane for the controlled LES case atF⁺= 1.5 (black & white). The mean flow is superimposed (colors). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.5. Turbulence model

For LES, the subgrid scale model is the selective mixed scale model proposed by Sagaut[16]. For URANS, the Spalart–Allmaras model[17], thek–xmodel of Wilcox[18], thek–xmodel of Men-ter with SST correction[19]and ak–lvariant of the EARSM model [20]have been assessed. The three first models are currently used in the aeronautic industry.

3. Results

3.1. A brief description of the flow using LES

Figs. 3 and 4show a numerical strioscopy of the instantaneous flow for the uncontrolled case and for a controlled case atF⁺= 1.5 respectively. Iso-contours of the mean flow are superimposed in order to quantify the global effect of the synthetic jet forcing. It can be observed in these figures that the flow is massively Fig. 5.Surface of the separated bubble non dimensionalized by the separated

bubble surface of the uncontrolled flow for different turbulence models.

Fig. 6.Surface of the separated bubble for the non controlled case (black solid line), the most efficient frequency (red dashed line) and the maximum frequency (blue dashed–dotted line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

separation. The strongest density gradients which are highlighted by strioscopy are present in the shear layer. An intense structure

Fig. 4. The caseF⁺= 1.5 is the one which maximizes the reduction of the separated bubble surface. The later is divided by about 2.8

Fig. 7.Fluctuating kinetic energy for the baseline base and for the best frequency for LES,k–xMenter, and EARSM models.

with respect to the one of the uncontrolled flow. The separation point remains nearly unchanged. Oppositely, the reattachment point decreases fromx/h= 4.7 tox/h= 3.7. This represents a reduc-tion of about 25% of the separareduc-tion length.

3.2. RANS/LES comparisons

The goal of the paper is to compare URANS and LES. The lat-ter is here considered as the reference since a successful valida-tion against DNS has already been performed in Ref. [1]. The chosen figure of merit to discriminate between models is the surface of the separated bubble. This quantity gives a more glo-bal overview of the control effect than the bubble length since very long thin bubble can sometimes be observed. It is defined as follows.

For each curvilinear coordinate (s,g), a local mass-flux between the wall and the ordinategtakes is computed as:

Qmðs;gÞ ¼

The separation bubble surface is thereby defined by the surface comprised between the wall and the contour C(s,g) where Qm(s,g) = 0.

Fig. 5presents the frequency response of the bubble surface non dimensionalized by the separated surface S0of the uncontrolled flow. Significant differences can be observed from one model to one another. Nevertheless, every model but the Spalart–Allmaras is shown to be sensitive to the periodic excitation of the synthetic jet. Bothk–xmodels and the EARSM one are able to identify a fre-quency of about 1.5 as being the optimal one consistently with LES results. The main limitations of URANS models are of two types.

First, the amplitude of the bubble surface reduction is at best for thek–xmodels equal to 40% whereas a reduction of 60% is regis-tered with LES. Secondly, the best RANS models already identified (k–x, EARSM) see almost no reduction of the bubble surface at high frequencies (F⁺> 4) whereas a 40% reduction is noticed with LES. URANS here behaves as a low pass filter, high frequency exci-tation being damped.

In order to offer a more quantitative view, the bounds of the separated bubble for the uncontrolled case, for the optimal fre-quency and for the highest frefre-quency is plotted in Fig. 6 both for URANS and LES. Maybe one of the most striking feature high-lighted by this figure is the fact that the uncontrolled flow is dif-ferent both between LES and URANS and between URANS models with each other. The separation length varies from x/

h= 5 to 6 for URANS whereas the value of 4.7 has been found for LES. For the uncontrolled flow, the best agreement with LES is obtained with the Spalart–Allmaras model. Unfortunately, the later model is almost insensitive to a frequency excitation.

As already mentioned,k–xmodels offer a correct frequency re-sponse at least for F⁺< 4. Nonetheless, the separation length is significantly overestimated for the uncontrolled flow. This is also true for the controlled flow at the optimal frequency. Overall, a good compromise may be the EARSM model which both repro-duces the uncontrolled flow reasonably well and responses cor-rectly to the periodic excitation at low frequency even if the bubble surface reduction is smaller than the ones of the k–x models. More complicated models like RSM ones might improve further the results. Even if the accuracy of URANS model appears to be insufficient to produce quantitative predictions, it is how-ever worthwhile to notice that from a design point of view it might be enough to use a tool able to identify the optimal fre-quency. The relevance of this remark is of course dependent of the type of control envisioned, high frequency forcing such as

the one used by Glezer et al.[11]may be inaccessible to URANS computation, LES or maybe RANS/LES coupling being the remain-ing expensive alternatives.

In order to improve the understanding of the models behavior, Fig. 7displays the fluctuating turbulent energy for the LES, thek–

xMenter model and the EARSM model both for the uncontrolled case and for the best controlled case. The two later models have been identified as giving the best results on the surface bubble reduction. The frequencies for which these three models are opti-mal are not exactly the same but it has been checked that a sopti-mall difference in the optimal frequencies does not bias the present analysis. Furthermore for URANS models, it is important here to make the distinction between the resolved kinetic energy and the modeled one. InFig. 7, the fluctuating kinetic energy which is the sum of the two is represented to be consistent with LES.

In the baseline case, the resolved kinetic energy is of course null for RANS models. At first sight, it can be observed that both UR-ANS models underestimate the turbulent kinetic energy in the baseline configuration. The error in the reattachment length is then attributed to an underestimation of the mixing and to a sub-sequent underestimation of the shear layer growth rate. In the controlled case, the level of unsteadiness which is essentially the result of the jet forcing is roughly the same for URANS models as for LES. In the actuator vicinity (in the range 0.5 <x/h< 1) the fluctuating kinetic energy reaches values as high as 3400 in each considered case. Further downstream in the LES case, the fluctu-ating turbulent energy decreases quickly and becomes lower than in the baseline case. Nevertheless, the fluctuation spectrum (not shown) evidences the fact that the forcing frequency remains dominant all along the separation bubble. In terms of length scales, a forcing at a frequency of about 1200–1500 Hz corre-sponds to a penetration length of abouth/2. In the LES computa-tion, these length scales grow. Unfortunately, the behavior of URANS models is physically different: downstream fromx/h= 2, the whole part of the resolved fluctuating energy has been con-verted into modelled kinetic energy. The length scales of the fluc-tuations which are initially the same as in the LES decrease drastically and downstream fromx/h= 2 the flow is in practice steady. It is then concluded that even a frequency as low as F= 1.5 is located in the dissipative range of the URANS modelling.

Trying to discriminate between thek–xand the EARSM model, it can be observed that the resolved kinetic energy is convected more downstream with the later than with the former model. It has been checked that the levels of turbulent viscosity are

Trying to discriminate between thek–xand the EARSM model, it can be observed that the resolved kinetic energy is convected more downstream with the later than with the former model. It has been checked that the levels of turbulent viscosity are