Tony Arts

This work focuses on numerical simulations of flows inblade internal cooling system. Large Eddy Simulation (LES) andReynolds-Averaged Navier Stokes (RANS) approaches are com-pared in a typical blade cooling related problem. The case isa straight rib-roughened channel with high blockage ratio, com-puted and compared for both a periodic and full spatial domains.The configuration was measured at the Von Karman Institute(VKI) using Particle Image Velocimetry (PIV) in near gas tur-bine operating conditions. Results show that RANS models usedfail to predict the full evolution of the flow within the channelswhere massive separation and large scale unsteady features areevidenced. In contrast LES succeeds in reproducing these com-plex flow motions and both mean and fluctuating components areclearly improved in the channels and in the near wall region.Periodic computations are gauged against the spatial computa-tional domain and results on the heat transfer problem are ad-dressed. NOMENCLATUREDh Hydraulic diameter (m)h Rib height (m)Nu Nusselt number : Nu = qwDhλ(Tw−Tre f )Re Reynolds number : Re = U0Dhν ∗Address all correspondence to this author.Pr Prandtl numberqw Wall heat flux (W.m−2)U0 Bulk velocity (m.s−1)Ux,Uy,Uz Streamwise, normal and spanwise components ofmean velocity (m.s−1)ux,uy,uz Streamwise, normal and spanwise components offluctuating velocity (m.s−1)ν Cinematic viscosity (m2.s−1)λ Thermal conductivity (W.m−2.K−1)Tw Wall temperature (K)Tre f Reference temperature (K)RANS Reynolds-Averaged Navier StokesLES Large Eddy SimulationDNS Direct Numerical SimulationNRMSD Normalized Root-Mean-Square Deviation INTRODUCTIONOutput power increase and improved efficiency of aeronau-tical engine can be obtained by increasing the combustor out-let temperature. This rise in temperature however imposes newconstraints on the turbine design since the blade material melt-ing point is often well surpassed. Life duration of the turbinecan hence greatly be reduced if the cooling system is poorlydesigned. Providing improved design tools and better predic-tions in this part of the engine becomes therefore more and morecritical. Today, design engineers heavily rely on computational 1Copyright c© 2012 by ASMEtel-00870685,version1-7Oct2013 FIGURE 1. EXAMPLE OF APPLICATION OF THIS STUDY :FLOW INSIDE A TURBINE BLADE (DASHED ARROW). tools to dimension these complex systems. For the fluid in themain vein and within the cooling ducts, RANS modeling is rou-tinely used and conjugated heat transfer problems are possible.Although RANS has known limitations for these two types offlows, it clearly benefits from a computational efficiency and themodeling experience gained in this specific context: 2D and 3Dcomputations (Ooi et al. [1]), specific modeling for wall boundedflows with low-Re turbulence model (Iacovides et Raisee [2] andunsteady computations with U-RANS methods (Saha. et al. [3]).However and with the advent of LES, a fully unsteady modelingapproach, improvement in the flow prediction can be expectedin this configuration, as already shown by Sewall et al. [4]. Fur-ther analysis of LES on ribbed channel configuration have alsobeen carried out, as for example the study of Tasti on the subgridstress [5], and in a more applied way the design study of Ahn andLee [6] with detached ribs.In this context, this article provides a comparison of RANSand LES computations on a cooling channel configuration repre-sentative of the industrial case of Fig. 1. A quick description ofRANS and LES modeling is first presented, then the ribbed chan-nel physical and numerical parameters are introduced to computeadiabatic and imposed heat flux operating point. The two lastsections show the results of these pure aerodynamic and aerother-mal computations, and compare them to 2D experimental fieldsof velocity and heat transfer enhancement factor. MODELISATION AND TOOLSIn the context of practical engineering calculations, the highReynolds numbers involved are too computationally expensiveto be simulated directly with Direct Numerical Simulation. Theinstantaneous (exact) governing equations need thus to be time-averaged, ensemble-averaged, or otherwise manipulated to re-move the small scales, resulting in a modifed set of equations thatare computationally less expensive to solve. However, the mod-ified equations contain additional unknown variables and turbu-lence models are needed to close the system before being sim-ulated numerically. The most common approach for complexconfigurations is still RANS that proposes to model the effect ofall the turbulent scales on the mean flow. An alternative and moreuniversal method is LES that introduces a separation between theresolved (large) turbulent scales and the modeled (small) scales(see Sagaut, Pope and Poinsot [7–9]). Both RANS and LESmethods, detailed below, have been heavily tested on aerody-namic setups and to a lesser extent on thermal problems. Forour work, two flow solvers are considered: a structured multi-block solver (elsA, see Cambier [10] for further description) andan unstructured solver (AVBP, detailed in the works of Mendezand Schonfeld [11, 12]).The governing equations for both approaches are the un-steady / steady compressible Navier-Stokes equations that de-scribe the conservation of mass, momentum and energy. In con-servative form, it can be expressed in three-dimensional coordi-nates as: dWdt+divF = 0(1) where W is the vector of primary variables, F = ( f − fv,g−gv,h− hv) is the flux tensor; f ,g,h are the inviscid fluxes andfv,gv,hv are the viscous fluxes. The fluid follows the idealgas law p = ρ r T , where r is the mixture gas constant. Thefluid viscosity follows Sutherland’s law and the heat flux followsFourier’s law. RANS approachThis modeling is one of the most used for engineering work.The principle is based for compressible flow on the Favre av-eraging of the Navier-Stokes equations, splitting each quantity(velocity, pressure...) into a Favre averaged and a fluctuatingcomponent: F = f̃ + f ′′ with f̃ =ρ fρ(2) With overline denoting Reynolds average. Substituting (2)into the Navier-Stokes equations (1) and averaging, one obtainsthe evolution equations to be solved numerically. Favre averag-ing introduces a new termτi j =−ρ ũ′′i u′′j in the momentum equa-tion, also called the Reynolds stresses. A common model em-ploys the Boussinesq hypothesis (see Schmitt [13]) based on themean strain rate tensor Si j. With this assumption, a new scalarνtalso known as the turbulent viscosity is introduced and is usuallyevaluated from available quantities. There are numerous types ofmodelling, classified in terms of number of transport equations 2Copyright c© 2012 by ASMEtel-00870685,version1-7Oct2013 solved (in addition to the RANS equations) to computeνt (asamong others works of Launder, Spalart and Durbin [14–17]).Here the k−L model of Smith [18] is used.The turbulent energy fluxqi present in the energy equationis modeled using a turbulent heat conductivity obtained from ν tby λ t = ρ̄ν tCp/Prt , where Prt = 0.7 is a constant turbulent Prandtlnumber, and the gradient of the Reynolds averaged temperaturecoming from the averaged state equation p=ρrT̃ : qi =−λt ∂ T̃∂xi(3) LES approachLES involves the spatial Favre filtering operation that re-duces for spatially, temporally invariant and localised filter func-tions to: ̃f (x, t) =1ρ(x, t)∫ +∞ −∞ρ(x′, t) f (x′, t)G(x′−x)dx′, (4) where G denotes the filter function. Within this specificcontext, the unresolved SGS stress tensorτi j is modeled usingthe Boussinesq assumption [19] in the same way as RANS butfrom filtered strain rate tensor S̃i j. The SGS turbulent viscosityis then computed in our case with the WALE model introducedby Nicoud [20], especially designed for wall bounded flow: νt =(Cw∆) (sijsdi j)3/2 (S̃i jS̃i j)5/2+(sijs di j)5/4,(5) sij =12 (g̃ij + g̃2ji)− 13 g̃kkδi j,(6) In Eq. (5), ∆ stands for the filter length (∝ the cubic-rootof the cell volume), Cw is the model constant equal to 0.4929and g̃i j is the resolved velocity gradient. The SGS energy flux ismodeled from the turbulent viscosity as for RANS method, withhere the filtered quantities [21–23]. Structured RANS solverThe elsA software uses a cell centered approach on struc-tured multiblock meshes. More information about this flowsolver are presented by Cambier [10]. Convective fluxes arecomputed with an upstream scheme based on the Advection Up-stream Splitting Method (AUSM) introduced by Liou [24]. Dif-fusive fluxes are computed with a second-order centered scheme.The pseudo time-marching is performed by using an efficientimplicit time integration scheme, based on the backward Eulerscheme and a scalar Lower-Upper (LU) Symmetric SuccessiveOver-Relaxation (SSOR) method as proposed by Yoon [25]. Theturbulent viscosity is computed with the two equations model ofSmith [18] based on a k−L formulation. Unstructured LES solverThe parallel LES code, AVBP [11, 12], solves the fullcompressible Navier-Stokes equations using finite elementschemes TTG4A and TTGC based on a two step TaylorGalerkin formulation for the convection [26] in a cell-vertexformulation [27, 28]. It is especially designed for LES on hybridmeshes. Indeed, these explicit low diffusion and low dispersionproperties provide 3rd order space and time accuracy. Employingexplicit schemes involves to keep the CFL number below 0.7to ensure stability, which lead to the main disadvantage to usetime steps very low: for example ∝ 10−7s for aerodynamicalapplications where the viscous sub-layer needs to be computedaccurately, inducing high computational costs. RIB-ROUGHENED COOLING CHANNEL CASEThe ribbed configuration used in this study, Fig. 2, is asquare channel with an hydraulic diameter Dh = 0.1m, an as-pect ratio of 1 and for which experimental measurements havebeen carried out at the Von Karman Institute. Square ribs normalto the flow are mounted on one wall with a pitch-to-height ratio(p/h) of 10 leading to a high-blockage ratio, h/Dh = 0.3, h be-ing the rib height. The Reynolds number of the mean flow basedon the bulk velocity and Dh is set to 40000, a regime similarof flows inside real cooling channels. Aerodynamic flow mea-surements have been performed using PIV by Casarsa [29] andthermal analysis using Liquid Crystal Thermography (LCT) byÇakan [30]. Mean flow velocity and fluctuations in the chan-nel and Enhancement Factor (EF) at the wall characterizing theheat-transfer are provided at the same operating point for com-parisons with numerical predictions. EF is the ratio between themeasured Nusselt number at the wall and Nu0, the Nusselt num-ber computed from the Dittus-Boelter correlation for a smoothcircular duct, as introduced by McAdams [31] : Nu0 = 0.023 ·Re0.8 ·Pr0.4(7) Computational domain and mesh generationMeasurements were performed in a section of the channelwhere the flow is assumed to be periodic. In order to reduce com-putational cost and allow finner mesh in LES, a first numerical 3Copyright c© 2012 by ASMEtel-00870685,version1-7Oct2013 FIGURE 2. COMPUTATIONAL DOMAIN OF THE RIBBEDCHANNEL domain is chosen to be only one section of the channel corre-sponding to one periodic pattern of the channel geometry, Fig. 2.In reality, turbine blade cooling channel may not reach a peri-odic regime due to a few number of ribs in one straight channel.A second case has thus been computed to analyse the periodicityhypothesis as a good approximation for a more realistic config-uration. The domain of study has been here chosen to be com-posed of 7 ribs and is called the spatial case. Periodic case Figure 2 gives a representation of the pe-riodic domain. Periodicity is applied between both flow inletand outlet sections. Since there is no inlet/outlet condition inthis case, fluid displacement is obtained by adding a source termsimilar to a fictive pressure gradient in the momentum equation,following the method of Cabrit [32]. All other boundaries arewalls following the adiabatic no-slip condition.Three periodic meshes are tested to see the influence of thediscretization on the LES results. The first one, called M1, is acoarse full-tetra mesh with low wall resolution and a small num-ber of cells allowing longer simulated physical time and fast sim-ulations, Fig. 3(a). The second mesh, M2, is a fine full-tetra meshwith better wall resolution and approximately ten times morecells than M1, Fig. 3(b). Finally, M3 is an hybrid mesh madeof one layer of prisms on all walls and tetrahedrons everywhereelse, Fig. 3(c). This mesh combines a relative small number ofcells with good wall resolution thanks to the use of prisms at thewall. Characteristics of these three meshes are summarized inTab. 1. Since RANS simulation requests less computation time,no periodic reduction is applied and a full channel with four ribsis discretized with 18.5 ·106 points and a y+ near 1 on walls. Spatial case The spatial geometry is made of 7 repeti-tions of the domain seen on Fig.2, lightly extended at both ex-(a)