COUPLING CONDITIONS FOR LES WITH DOWNSTREAM RANS FOR THE PREDICTION OF INCOMPRESSIBLE TURBULENT FLOWS

Transcription

1 COUPLING CONDITIONS FOR WITH DOWNSTREAM RANS FOR THE PREDICTION OF INCOMPRESSIBLE TURBULENT FLOWS Dominic von Terzi Institut für Hdromechanik, Universität Karlsruhe Kaiserstraße, D-76 Karlsruhe, German Jochen Fröhlich Lehrstuhl für Strömungsmechanik, Technische Universität Dresden George-Bähr-Straße c, D-69 Dresden, German ABSTRACT Interfacing Large Edd Simulation () with a downstream Renolds-Averaged Navier Stokes (RANS) zone for incomressible flows is investigated. The mean velocit fields are matched at the redefined interface and velocit fluctuations of the zone are allowed to leave the domain b emloing a convective boundar condition. For incomressible flows, in addition, it is also necessar to rescribe conditions for the ressure or an equivalent variable. Two situations are considered: (i) The instantaneous ressure is solved globall within the union of the and RANS domains and (ii) the ressure fields are comletel decouled. For the latter, several alternatives of handling the issue of mass conservation at the interface were studied. The erformance of the different methods is scrutinized for turbulent channel flow and the flow over eriodic hills. It was found that for the first test case the ressure couling was uncritical whereas, for the hill flow, decouling of the ressure with exlicitl enforced mass conservation at the interface ielded the best and, for some situations, the onl accetable results. INTRODUCTION Renolds-Averaged Navier Stokes (RANS) calculations are able to deliver reliable results for man flows encountered in alications of engineering interest. In situations where these are not sufficient, Large Edd Simulation () is the next best choice. For, the large-scale motion is comuted directl and onl the small-scale motion is modeled, hence information on large coherent structures can be gained and less strict modeling assumtions need to hold. The rice to a is a substantiall higher comutational cost. In fact, it ma be so high that is not affordable for a ver comlex high-renolds number flow or an extensive arameter stud. A remed is the couling of and RANS calculations. Such a hbrid method restricts the use of the more exensive to regions of the flow field where RANS redictions are likel to be inadequate, e.g. in regions where the effects of large coherent structures are of interest. The interfacing of with a downstream RANS zone considered here can be ver instructive as an intermediate ste towards wall-modeling for, but it is also of ractical interest in its own right. This is the case for flows where changes in the mean ressure field and/or geometr far downstream of the region of rimar interest have a sufficientl strong ustream influence. For such kind of flows, a simle outflow condition cannot be alied close to the region of interest as was shown for a swirl stabilized model combustor b Pierce & Moin (998). However, lengthening the domain b adding a RANS zone and hence including the downstream effects ma be an attractive alternative to more comlex boundar conditions (von Terzi et al., 5). VELOCITY COUPLING For -to-rans te boundaries in flows with stationar statistics, the RANS zone can and should onl rovide mean values, whereas the delivers a time-deendent solution. Therefore, the role of the interface is to allow for mean flow information to roagate ustream and for the fluctuations to leave the domain without reflections. To this end, a general and arameter-free method can be alied (von Terzi, Fröhlich & Mar, 6) that coules the exlicitl Renolds-averaged variable at the outflow directl to the RANS inflow boundar, whereas fluctuations are convected out of the domain using a one-dimensional, linear convection equation: φ t + Un φ n = () where n is the direction normal to the interface, U n = u n the averaged velocit in the n-direction, and φ the resolved fluctuations of the quantit to be couled. For incomressible flow, onl the resolved velocities are couled this wa, hence φ = u i = u i u i. Here, the overbar denotes the filter imlied b and the brackets reresent the Renolds average which is exlicitl alied in an homogeneous satial direction of the interface lane and in time. The hsical meaning of () is that the downstream transort of fluctuations across the interface is dominated b convection. For this to be true, U n > and U n u n () In addition, laminar and modeled turbulent diffusion across the interface must be negligible which, however, is uncritical for turbulent flows and adequatel resolved. The couling condition of () is imlemented in its discrete form using a first-order uwind difference in the n-direction (index j along a grid line normal to the interface) and a so-called θ scheme (Tannehill et al., 997) with

2 θ in time (t = m t): φ m+ j = C φ m j + C φ m j + C φ m+ j () C = + θ Ũn ) C = ( ( θ) Ũn / C ( ) C = ( θ) Ũn / C C = θ Ũn/ C where, for a Finite Volume method, the interface is located at the face between the cells with index j (RANS-side) and j (-side). For θ =.5, as chosen for all simulations resented here, this results in the imlicit second-order accurate traezoid rule. The arameter Ũ n = U n t/ n () is the Courant Friedrichs Lew (CFL) number for the convective roblem. The resulting couling conditions are then U m+ j = um+ j (5) for the streamwise velocit U at the inflow boundar of the RANS calculation and u m+ j = U m+ j + u m+ j (6) for the resolved streamwise velocit u at the outflow boundar, with u obtained from () using a lower bound for Ũn of to ensure U n >. All other velocit comonents are comuted accordingl. Setting C = C = and C =.98 in (), recovers the ad hoc formula for the so-called enrichment strateg of Quéméré & Sagaut (). In addition to the downstream RANS considered here, the successfull alied this technique also as a wall-model for b lacing the RANS zone between the zone and the wall. PRESSURE COUPLING For incomressible flows with well-osed boundar conditions, mass conservation inside the fluid domain is imlicitl enforced through the ressure field or an equivalent constraint variable, e.g. the streamfunction. These variables are governed b a Poisson-te equation such that the convective condition () cannot be alied to their fluctuations and a different wa of couling the and RANS domains needs to be devised. In the following we restrict ourselves to formulations involving the ressure. Two distinct ossibilities of handling this variable at the interface are scrutinized. One ossibilit is to solve the instantaneous ressure globall in the union of the and the RANS domains. We denote this case C in order to kee the notation of von Terzi et al. (6). It is a strong couling that enforces instant mass conservation in the comlete fluid domain. If the algorithm for the Poisson solver emloed uses a domain decomosition technique no adjustment to the algorithm is necessar making this a ver attractive aroach. However, some turbulence models, in articular most of the subgridscale models emloed for, reresent onl the deviatoric art of the unresolved stresses. As a consequence, a common interretation of the ressure in the corresonding momentum equations is that of a modified ressure, where the missing isotroic term of the unresolved stresses is imlicitl added to the hsical ressure. If this is the case on an side of the RANS interface, then the sudden change in hsical meaning of a ressure solved continuousl ma cause comlications. An alternative aroach is to decoule the ressure fields of the and RANS domains comletel. In this case, both the velocit and ressure fields are discontinuous at the interface and mass conservation across this boundar is not guaranteed. Ignoring the mass conservation issue is the method identified as case P below. P constitutes a weak couling since b matching the averaged velocities at the interface, the mean mass flux is maintained. Instantaneousl, however, this is true onl if the net mass transort of the fluctuations across the interface sums u to zero at each instant in time. This is likel not to be the case at startu of a simulation, when the exlicit time average has not converged et, and for situations where () is violated at the interface, e.g. close to walls and for reverse flow. As a consequence of a lack of instantaneous mass conservation, the boundar conditions for the Poisson solver become ill-osed and the solver converges oorl or not at all. The roblem of an integral mass flux imbalance is not limited to /RANS couling, but occurs routinel with rojection methods for incomressible flow. As a remed, a global mass flux correction is alied comuting the mass flux over all inflow boundaries ṁ in and calculating the actual mass flux over all exit boundaries ṁ out due to the uncorrected exit velocities u i. All velocit comonents at all outflow cells can then be scaled with the same factor u i = f m u i with f m = ṁ in ṁ out (7) In ractical simulations, the mass flux ratio f m in (7) is usuall ver close to one. This global correction can also be alied to the velocities constructed with (5) and (6) at the /RANS interface. For the boundar cells, one needs to relace ṁ out in (7) with the mass flux leaving the domain ṁ. Conversel, the velocities in the RANS boundar cell are then scaled using the magnitude of the mass flux entering the RANS domain ṁ RANS. The simulation with decouled ressure and a global mass flux correction is called case P below. For couling of and RANS in case of comlex geometries, we might have multile embedded domains making the global correction cumbersome, if not imossible. Hence, a local aroximation to the global flux correction above is roosed here with ṁ in in (7) being relaced, at each siml connected interface, b ṁ interface = / (ṁ + ṁ RANS ) (8) with ṁ and ṁ RANS determined b integration over the corresonding interface. A simulation using this local correction is denoted as case P. TURBULENCE MODELS AND NUMERICAL METHOD For all cases resented here, the Smagorinsk model and the Salart Allmaras model are used for the and RANS regions, resectivel. The constant in the Smagorinsk model was chosen as C s =.65. For the RANS model, a transort equation for a modified turbulent viscosit ν t is solved. This equation requires a reasonable inlet-te boundar condition at the interface. It is rovided b solving the transort equation also in the domain, albeit using the exlicitl Renolds-averaged velocities. The simulations were erformed with the Finite Volume flow solver OCC (Hinterberger, ) develoed at the

3 Institute for Hdromechanics at the Universit of Karlsruhe. It solves the incomressible time-deendent filtered Navier Stokes equations together with transort equations required for the turbulence models on bod-fitted curvilinear blockstructured grids. Second-order central differences are used for the discretization of the convection and diffusion fluxes. Onl for the convection term in the transort equation for the edd viscosit the monotonic HLPA scheme is emloed. An exlicit, low-storage, three-stage Runge Kutta method is used for time advancement. TURBULENT CHANNEL FLOW First, we will comare the different ressure couling techniques (C and P ) for full develoed turbulent channel flow at a Renolds number based on the friction velocit and channel half-width of Re τ = 95 and based on bulk velocit of Re = 7. For this setu, there exist Direct Numerical Simulation (DNS) data from Moser et al. (999) for comarison and an adverse effect of a couling can be immediatel detected in form of streamwise variations of statistics or reflections of instantaneous quantities (von Terzi et al., 6). The domain is divided into three arts: the inflow, the rincial three-dimensional zone and the two-dimensional RANS zone. All quantities are made dimensionless using the channel half-height δ and the bulk velocit U b at the inlet of the rincial zone. The inflow is a stand-alone with eriodic boundar conditions in the streamwise direction and a mass flux enforced b volume forces. It rovides lanes of instantaneous velocities for the inflow of the rincial zone. For each of the zones, the domain size is π π in the streamwise (x), wall-normal (), and sanwise (z) directions, resectivel. Grid stretching is emloed onl in the wall-normal direction. Both domains are discretized using 8 8 cells resulting in a near-wall scaling of + =.5, x+ =, and z + = 6. The RANS domain extends over a length of π on a stretched grid in the streamwise direction. The same wall-normal grid as for the zones is utilized with one cell in the sanwise direction. The time ste was t =. and statistics were samled over t = 5 δ/u b starting after stead statistics were obtained. All averages were taken in time and the lateral direction. In figure (a) and (b), wall-normal rofiles of mean streamwise velocit and resolved longitudinal Renolds stress in near-wall coordinates are shown. The rofiles were taken from the main domain in the lane adjacent to the interface. All simulations comare well to the DNS reference data with the couled simulations exhibiting a slight imrovement over the data comuted on the same grid without couling. Figure (c) demonstrates the change in meaning of the ressure between DNS,, and RANS. Integrating the averaged wall-normal momentum equation results in + v v = const. as seen for the DNS data. Hence, the mean ressure of the DNS and the mean modified ressure of the change with. On the other hand, consistent with its modeling assumtions, the Salart Allmaras model ields a constant modified ressure in the wall-normal direction, i.e. RANS = + K/ = const., where K is the turbulent kinetic energ. However, K/ v v due to the near-wall anisotro. As a consequence, close to the wall, should differ from RANS across the interface and some adjustment is to be exected. A simle correction matching RANS and + K res / with K res being the resolved turbulent kinetic energ is not adequate, whereas using + v v is not general enough to hold for other flow configurations. Figure rovides a closer look at the interface. The streamwise develoment of instantaneous velocit and ressure and the mean ressure at three different wall-normal locations are shown for cases C, P, and P. The results of P are ver similar to the other P-cases and are therefore not shown for brevit. The lots demonstrate that the instantaneous velocities fluctuate in the inflow, move on through the rincile domain and leave the latter without reflections. On the RANS side of the interface, a stead mean velocit is obtained. A jum at the interface can be discerned since onl the mean values are continuous. The instantaneous ressure behaves alike, albeit with two differences: One is that, in the inflow, the mean ressure gradient is missing, since it is reresented b the volume forces instead. The other is that for the global ressure couling (C) the ressure is continuous at the interface whereas for the P-cases a jum similar to the velocities is clearl visible, but the mean flow gradient is continuous. For case C, in the near-wall region of the interface, a slight adjustment of the mean ressure can be seen. This is due to the change in modified ressure as discussed above. Since cases P and P agree well with case P, mass flux corrections are obviousl uncritical for this configuration. In fact, f m was of the order of, for both P and P. The global mass flux correction (P) achieved the best convergence rate of the Poisson solver, whereas the maximum residual of case P remained one order of magnitude larger than for all other cases ( 5 instead of 6 ). Contrar to variations in the velocit couling (von Terzi et al., 6), the different ressure interfacing methods tested here all delivered excellent results and no clear sueriorit of an of these methods can be discerned. FLOW OVER PERIODIC HILLS The channel flow is a sensitive but uncritical test case since the flow is full develoed so that no downstream information is reall needed for the ustream. This is different for the flow over eriodic hills. This configuration was devised b Mellen et al. () and detailed reference data were resented b Fröhlich et al. (5). The Renolds number based on the hill height h and the bulk velocit over the crest is Re b = 595. The distance from hill to hill is 9h, the domain height.6h, and the width.5h. As with the channel flow, the simulation is divided into three distinct zones (figure ). The first zone is comuted with using wall functions and eriodic boundar conditions in the downstream direction. 6 9 cells are used in the downstream, wall-normal and lateral directions, resectivel. This zone rovides reference and inflow data. For the second zone, is erformed using the same resolution as in Zone. At a re-defined location, the simulation switches from to RANS, and the third zone with a twodimensional grid begins. At the RANS outflow, Neumann boundar conditions are alied. t and the total simulation time was roughl 9 h/u b. Two locations for the /RANS interface are considered: (i) on the crest of the hill and (ii) before the crest of the hill as illustrated in figure. For the interface laced on the crest, all methods ielded excellent results making this an uncritical test. The interface at x 7, on the other hand, constitutes a severe challenge. A conventional convective outflow condition at this osition,

4 i.e. no couling to a RANS calculation, roduced strong reflections and regions of searation before the outflow lane, before eventuall the solution diverged. The same haened for cases C and P. This could have been exected, in articular, considering the strong fluctuations of large coherent structures in the center of the channel and the slight searation bubble at the bottom of the ustream hill front. Both are likel to cause violations of the assumtions for the velocit couling stated in (). Nonetheless, cases P and P, i.e. the decouled ressure fields with mass flux correction, ielded reasonable results. This is shown in figure for case P: The mean streamlines reveal that, for the two-dimensional RANS solution, reattachment occurs far too late, consistent with RANS results in the literature showing that this method is inadequate for redicting the searated flow region. However, the RANS zone successfull rovides the ustream domain with the ressure and mean velocit information due to the streamline curvature induced b the hill rising behind the interface. Therefore, the in Zone is able to deliver results similar to the reference solution of Zone, albeit with a slightl longer recirculation region. Both reattachment lengths of. h and. h for Zone and, resectivel, are close to the reference value of.6 to.7 h of Fröhlich et al. (5). The instantaneous velocit contours show that the RANS flow field is indeed stead, whereas the unstead velocities on the side of the interface leave the domain without an obvious reflections. More quantitativel, in figure, rofiles of mean streamwise velocit and resolved Renolds stresses obtained from simulations P and P, ure on the same grid, and a reference of Breuer & Jaffrézic (5) emloing twelve million cells are comared for selected downstream locations. The region of rimar interest for the hill flow is arguabl the recirculation region as shown in lots (a) and (d) (f). Differences to the reference can be exected for the mean flow and must be exected for the resolved fluctuations due to the coarser resolution, but these are tolerabl small. Differences between the couled simulations P and P comared to the stand-alone cannot be discerned for the mean flow and are small for the resolved stresses where the constitute a slight imrovement. Even in close roximit to the interface mean flow deviations remain small (figure b). The temoral sectrum of the streamwise velocit at a oint in the domain right at the interface is shown in figure (c). No artificial eaks in the sectrum aear which might be generated b the resence of the interface. The comare well to the reference sectra at nearb locations of Fröhlich et al. (5). Other than for the channel flow simulations, the flux correction is crucial for the hill flow and was substantiall larger (f m = O( )). Overall, the convective velocit couling with decouled ressure fields and mass flux correction achieved the main objective of an /RANS couling: the region was shortened without deterioration of the results in regions of interest. The rear of the hill was then comuted using a two-dimensional RANS calculation, i.e. a fraction of the number of cells in this region of lesser interest was emloed. CONCLUSION Couling of with a downstream RANS at a shar interface was investigated for incomressible flows. On the -side of the interface a convective condition (von Terzi et al., 6) was alied to the velocit fluctuations whereas the exlicitl averaged velocit was directl couled to its counterart in the RANS domain. This method can be regarded as a generalization of the enrichment technique of Quéméré & Sagaut (). It is free of arameters and more robust. The method erformed well for turbulent channel flow irresective of the choice of ressure couling technique. Deending on the lacement of the RANS interface, the hill flow is a challenging test case for couling to a downstream RANS calculation. It was shown that the resent velocit couling can ield ver good results. For the ressure, it turned out to be necessar to decoule the two domains comletel and then to exlicitl enforce mass conservation across the interface. This was achieved b scaling the total velocities on either side. The mass flux to be enforced was either secified globall, e.g. as the mass flux through all inlets of the domain, or determined locall b averaging the existing fluxes on both sides of the interface. The local mass flux correction is more general, easier to imlement for comlex flow situations, and seems to be ver robust. Although this method requires a long startu time until the running averages are sufficientl converged and is less strict in enforcing continuit, decouling of ressure with a local mass flux correction delivered ver romising results for both test cases considered. Taking into account that onl the decouled ressure with mass flux correction delivered accetable results for all configurations, it is recommended as the standard incomressible comlement to the convective velocit couling. REFERENCES Breuer, M. & Jaffrézic, B., 5, New reference data for the hill flow test case. htt:// CNRS/. Fröhlich, J., Mellen, C. P., Rodi, W., Temmerman, L. & Leschziner, M. A., 5, Highl resolved large-edd simulation of searated flow in a channel with streamwise eriodic constrictions, J. Fluid Mech. 56, Hinterberger, C.,, Dreidimensionale und tiefengemittelte Large-Edd-Simulation von Flachwasserströmungen, PhD thesis, Universit of Karlsruhe. Mellen, C. P., Fröhlich, J. & Rodi, W.,, Large edd simulation of the flow over eriodic hills, Proceedings, 6th IMACS World Congress. Lausanne, Switzerland. Moser, R. D., Kim, J. & Mansour, N. N., 999, Direct numerical simulation of turbulent channel flow u to Re τ = 59, Phs. Fluids, Pierce, C. D. & Moin, P., 998, Large Edd Simulation of a confined coaxial jet with swirl and heat release, AIAA aer Quéméré, P. & Sagaut, P.,, Zonal multi-domain RANS/ simulations of turbulent flows, Int. J. Num. Meth. Fluids, Tannehill, J. C., Anderson, D. A. & Pletcher, R. H., ed. 997, Comutational Fluid Mechanics and Heat Transfer, nd edn., Cha...5. Talor & Francis. von Terzi, D. A., Fröhlich, J. & Mar, I., 6, Zonal couling of with downstream RANS calculations, Submitted to Theoret. Comut. Fluid Dnamics. von Terzi, D. A., Hinterberger, C., García-Villalba, M., Fröhlich, J., Rodi, W. & Mar, I., 5, with downstream RANS for flow over eriodic hills and a model combustor flow, EUROMECH Colloquium 69, Large Edd Simulation of Comlex Flows, TU Dresden, German, Oct. 6 8, 5.

5 <u> + 5 DNS C P P P <u u > DNS C P P P DNS: <> + DNS: <> + + K + / DNS: <> + + <v v > + : < > + : < > + + K res,+ / : < > + + <v v > + RANS: RANS, (a) mean streamwise velocit (b) resolved longitudinal Renolds stress (c) ressure and modified ressure Figure : Turbulent channel flow: Comarison of results with DNS reference data in near-wall scaling and role of modified ressure; rofiles for cases C, P, P, and P have been obtained at the grid line next to the interface, denotes data from the inflow. Inflow. Inflow. Inflow... u (a) C: instantaneous streamwise velocit (b) C: instantaneous ressure (c) C: mean ressure Inflow. Inflow. Inflow... u (d) P: instantaneous streamwise velocit (e) P: instantaneous ressure (f) P: mean ressure Inflow. Inflow. Inflow... u (g) P: instantaneous streamwise velocit (h) P: instantaneous ressure (i) P: mean ressure Figure : Turbulent channel flow simulations with globall couled (C) and decouled (P and P) ressure fields. Streamwise rofiles at different wall-normal locations from to to bottom: center of channel, =. ( + = ), and =.7 ( + =.5); arbitrar offset added to ressure for clarit.

6 OUTLET 6 5 D Grid D Grid ZONE = eriodic INLET ZONE = ZONE = RANS (Neumann boundar) x (a) mean streamlines 6 5 D Grid D Grid ZONE = eriodic INLET ZONE = ZONE = RANS OUTLET (Neumann boundar) x Level U-velocit: (b) instantaneous streamwise velocit Figure : Flow over eriodic hills with interface at x 7: Setu of simulation and results for case P. reference P P reference P P E f -5/ P P <u> <u> -8 - f (a) x = (b) x = 7 (c) temoral sectrum at x = 7, = reference P P reference P P reference P P <u u > <v v > <u v > (d) resolved longitudinal Renolds stress (e) resolved wall-normal Renolds stress (f) resolved Renolds shear stress Figure : Flow over eriodic hills with interface at x 7: Profiles of mean streamwise velocit at selected locations, ower sectrum densit of streamwise velocit at the interface, and resolved Renolds stresses at x = ; reference from Breuer & Jaffrézic (5); denotes data from the inflow.