Numerical study of oscillating boundary layer flow over a flat plate using k–kL–ω turbulence model

Oscillating boundary layer flow over an infinite flat plate at rest was simulated using the k–k_L–ω turbulence model for a Reynolds number range of 32 ⩽ Re_δ ⩽ 10,000 ranging from fully laminar flow to fully turbulent flow. The k–k_L–ω model was validated by comparing the predictions with LES results and experimental results for intermittently turbulent and fully turbulent flow regimes. The good agreement obtained between the k–k_L–ω model prediction with the experimental and LES results indicate that the k–k_L–ω model is able to accurately simulate transient intermittently turbulent flow and as well as accurately predict the onset of turbulence for such oscillatory flows.     

Supersonic flow past a system of two swept wedges mounted on a preliminary compression surface

The results of a numerical modeling of flow past a configuration consisting of two wedges with swept leading edges, so mounted on a preliminary compression surface that the beveled wedge surfaces deflect the wedge-compressed flows counter to each other, are presented. The calculations are performed on the basis of the averaged Navier-Stokes equations, together with the SST k-ω turbulence model, at the freestream Mach number M = 6. For the configuration geometry chosen the flow pattern is characterized by an irregular interaction between the wedge-induced shocks in the plane of symmetry. These shocks also induce three-dimensional, quasi-conical separations of a turbulent boundary layer on the preliminary compression wedge. In the separation zones the flows are directed toward the plane of symmetry of the configuration and interact with one another with the formation of a typical central “bulged” separation flow zone.     

An implicit upwind parabolized Navier-Stokes code for chemically nonequilibrium flows

The previously developed single-sweep parabolized Navier-Stokes (SSPNS) space marching code for ideal gas flows has been extended to compute chemically nonequilibrium flows. In the code, the strongly coupled set of gas dynamics, species conservation, and turbulence equations is integrated with the implicit lower-upper symmetric Gauss-Seidel (LU-SGS) method in the streamwise direction in a space marching manner. The AUSMPW+ scheme is used to calculate the inviscid fluxes in the crossflow direction, while the conventional central scheme for the viscous fluxes. The k-g two-equation turbulence model is used. The revised SSPNS code is validated by computing the Burrows-Kurkov non-premixed H2/air supersonic combustion flows, premixed H2/air hypersonic combustion flows in a three-dimensional duct with a 15° compression ramp, as well as the hypersonic laminar chemically nonequilibrium air flows around two 10° half-angle cones. The results of these calculations are in good agreement with those of experiments, NASA UPS or Prabhu's PNS codes. It can be concluded that the SSPNS code is highly efficient for steady supersonic/ hypersonic chemically reaction flows when there is no large streamwise separation.     

An immersed boundary/wall modeling method for RANS simulation of compressible turbulent flows

In this paper, an immersed boundary (IB) method is developed to simulate compressible turbulent flows governed by the Reynolds‐averaged Navier‐Stokes equations. The flow variables at the IB nodes (interior nodes in the immediate vicinity of the solid wall) are evaluated via linear interpolation in the normal direction to close the discrete form of the governing equations. An adaptive wall function and a 2‐layer wall model are introduced to reduce the near‐wall mesh density required by the high resolution of the turbulent boundary layers. The wall shear stress modified by the wall modeling technique and the no‐penetration condition are enforced to evaluate the velocity at an IB node. The pressure and temperature at an IB node are obtained via the local simplified momentum equation and the Crocco‐Busemann relation, respectively. The SST k ? ω and S‐A turbulence models are adopted in the framework of the present IB approach. For the Shear‐Stress Transport (SST) k ? ω model, analytical solutions in near‐wall region are utilized to enforce the boundary conditions of the turbulence equations and evaluate the turbulence variables at an IB node. For the S‐A model, the turbulence variable at an IB node is calculated by using the near‐wall profile of the eddy viscosity. In order to validate the present IB approach, numerical experiments for compressible turbulent flows over stationary and moving bodies have been performed. The predictions show good agreements with the referenced experimental data and numerical results.     

Hybrid RANS/LES computations of plane impinging jets with DES and PANS models

The qualities of a DES (Detached Eddy Simulation) and a PANS (Partially-Averaged Navier–Stokes) hybrid RANS/LES model, both based on the k–ω RANS turbulence model of Wilcox (2008, “Formulation of the k–ω turbulence model revisited” AIAA J., 46: 2823–2838), are analysed for simulation of plane impinging jets at a high nozzle-plate distance (H/B = 10, Re = 13,500; H is nozzle-plate distance, B is slot width; Reynolds number based on slot width and maximum velocity at nozzle exit) and a low nozzle-plate distance (H/B = 4, Re = 20,000). The mean velocity field, fluctuating velocity components, Reynolds stresses and skin friction at the impingement plate are compared with experimental data and LES (Large Eddy Simulation) results. The k–ω DES model is a double substitution type, following Davidson and Peng (2003, “Hybrid LES–RANS modelling: a one-equation SGS model combined with a k–ω model for predicting recirculating flows” Int. J. Numer. Meth. Fluids, 43: 1003–1018). This means that the turbulent length scale is replaced by the grid size in the destruction term of the k-equation and in the eddy viscosity formula. The k–ω PANS model is derived following Girimaji (2006, “Partially-Averaged Navier–Stokes model for turbulence: a Reynolds-Averaged Navier–Stokes to Direct Numerical Simulation bridging method” J. Appl. Mech., 73: 413–421). The turbulent length scale in the PANS model is constructed from the total turbulent kinetic energy and the sub-filter dissipation rate. Both hybrid models change between RANS (Reynolds-Averaged Navier–Stokes) and LES based on the cube root of the cell volume. The hybrid techniques, in contrast to RANS, are able to reproduce the turbulent flow dynamics in the shear layers of the impacting jet. The change from RANS to LES is much slower however for the PANS model than for the DES model on fine enough grids. This delays the break-up process of the vortices generated in the shear layers with as a consequence that the DES model produces better results than the PANS model.     

A modified STRUCT model for efficient engineering computations of turbulent flows in hydro-energy machinery

A modified STRUCT (MST) turbulence model for efficient engineering computations of turbulent flows in hydro-energy machinery is proposed in this paper. The MST model switches between URANS and LES-like modes using a new damping function to adjust the turbulent viscosity. Compared with the original STRUCT method, the modifications are as follows: (1) the BSL k-ω model with the Spalart-Shur correction is chosen as the new baseline to improve the sensitivity to rotation and curvature; (2) a new adaptive time-scale ratio is proposed to avoid the arbitrariness of geometric averaging operation in the original method; (3) the normalized helicity is introduced into the new damping function to detect the energy backscatter phenomenon. Five classical high Reynolds number flow cases are tested. The results show that the turbulent viscosity of the MST model is reasonably reduced in the massively separated regions and LES-like mode is activated, which captures more turbulent vortices and fluctuations on the URANS grids. With high efficiency and robustness, the MST model inherits the advantages of the original STRUCT method and improves the prediction accuracy of the turbulence with rotation and curvature, which enables efficient engineering computations of turbulent flows in hydro-energy machinery.     

Exploring k and ? with R–Equation Model Using Elliptic Relaxation Function

An extended version of the isotropic R?Cequation model accompanied by an elliptic relaxation approach to account for the distinct effects of low-Reynolds number (LRN) and wall proximity is proposed. The turbulent kinetic energy k and the dissipation rate ? are evaluated using the R ( $=k^2/\tilde{\epsilon}$ ) transport equation together with some empirical relations. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and the Schwarz?? inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence in the sense that they are sensitized to rotational and nonequilibrium flows. The model is validated against a few well-documented flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Comparisons indicate that the present model offers some improvement over the Spalart?CAllmaras one?Cequation model and competitiveness with the SST k?C?? model.     

A modular RANS approach for modelling laminar–turbulent transition in turbomachinery flows

In this study we propose a laminar–turbulent transition model, which considers the effects of the various instability modes that exist in turbomachinery flows. This model is based on a K–ω–γ three-equation eddy-viscosity concept with K representing the fluctuating kinetic energy, ω the specific dissipation rate and γ the intermittency factor. As usual, the local mechanics by which the freestream disturbances penetrate into the laminar boundary layer, namely convection and viscous diffusion, are described by the transport equations. However, as a novel feature, the non-local effects due to pressure diffusion are additionally represented by an elliptic formulation. Such an approach allows the present model to respond accurately to freestream turbulence intensity properly and to predict both long and short bubble lengths well. The success in its application to a 3-D cascade indicates that the mixed-mode transition scenario indeed benefits from such a modular prediction approach, which embodies current conceptual understanding of the transition process.     

Hybrid RANS/LES of flow and heat transfer in round impinging jets

Fluid flow and convective heat transfer predictions are presented of round impinging jets for several combinations of nozzle-plate distances H/D = 2, 6 and 13.5 (where D is the nozzle diameter) and Reynolds numbers Re = 5000, 23,000 and 70,000 with the newest version of the k-ω model of Wilcox (2008) and three hybrid RANS/LES models. In the RANS mode of the hybrid RANS/LES models, the k-ω model is recovered. Three formulations are considered to activate the LES mode. The first model is similar to the hybrid models of Davidson and Peng (2003) and Kok et al. (2004). The turbulent length scale is replaced by the grid size in the destruction term of the k-equation and in the definition of the RANS eddy viscosity. As grid size, a maximum measure of the hexahedral grid cell is used. The second model has the same k-equation, but the eddy viscosity is the minimum of the k-ω eddy viscosity and the Smagorinsky eddy viscosity, following a proposal by Batten et al. (2004). The Smagorinsky eddy viscosity is formed with the cube root of the cell volume. The third model has, again, the same k-equation, but has an eddy viscosity which is an intermediate between the eddy viscosities of the first and second models. This is reached by using the cube root of the cell volume in the eddy viscosity formula of the first model.The simulation results are compared with experimental data for the high Reynolds number cases Re = 23,000 and Re = 70,000 and LES data for the low-Reynolds number case Re = 5000. The Reynolds numbers are defined with the nozzle diameter and the bulk velocity at nozzle outlet. At low nozzle-plate distance (the impingement plate is in the core of the jet), turbulent kinetic energy is overpredicted by RANS in the stagnation flow region. This leads to overprediction of the heat transfer rate along the impingement plate in the impact zone. At high nozzle-plate distance (the impingement plate is in the mixed-out region of the jet), the turbulence mixing is underpredicted by RANS in the shear layer of the jet which gives a too high length of the jet core. This also results in overprediction of the heat transfer rate in the impingement zone caused by too big temperature gradients at impingement.All hybrid RANS/LES models are able to correct the heat transfer overprediction of the RANS model. For good predictions at low nozzle-plate distance, it is necessary to sufficiently resolve the formation and development of the near-wall vortices in the jet impingement region. At high nozzle-plate distance, the essence is to capture the evolution and breakup of the flow unsteadiness in the shear layer of the jet, so that accurate mean and fluctuating velocity profiles are obtained in the impingement region. Although the models have a quite different theoretical justification and generate a quite different eddy viscosity in some flow regions, their overall results are very comparable. The reason is that in zones that are crucial for the results, the models behave similarly.     

Suitability of the k–ω turbulence model for scramjet flowfield simulations

The suitability of Wilcox's 2006 k– ω turbulence model for scramjet flowfield simulations is demonstrated by validation against five test cases that have flowfields representative of those to be expected in scramjets. The five test cases include a 2D flat plate, an axisymmetric cylinder, a backward‐facing step, the mixing of a pair of coaxial jets and the interaction between a shock wave and turbulent boundary layer. A generally good agreement between the numerical and experimental results is obtained for all test cases. These tests reveal that despite the turbulence model's sensitivity to freestream turbulence properties, the numerically predicted skin friction agrees with experimental data and theoretical correlations to their degree of uncertainty. The tests also confirm the importance of using a y⁺ value of less than 1 in getting accurate surface heat transfer distributions. In the coaxial jets case, the importance of matching the turbulence intensities at the inflow plane in improving the predictions of the turbulent mixing phenomena is also shown. A review of guidelines with regard to the setting up of grids and specification of freestream turbulence properties for turbulent Reynolds‐averaged Navier–Stokes CFD simulations is also included in this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulation of the thermal method for reducing turbulent friction

The results of calculating a supersonic turbulent boundary layer on a heated surface on the basis of the algebraic two-parameter (k-ε) and four-parameter (k-ε-θ ²-ε ₆) models of turbulence are compared with experimental data. Emphasis is placed on the ability of the models to predict the behavior of the friction and heat-transfer coefficients on a heated surface. The optimal model of turbulence is chosen. The possibility of improving the efficiency of viscous drag reduction by localizing the regions of heat addition to the boundary layer is demonstrated on the basis of numerical calculations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–68, January–February, 1998. This research was carried out with financial support from the International Scientific and Technological Center (project No. 199).     

Modeling the Interaction of a Vortex Pair with the Ground

The results of a numerical investigation of the interaction between an aircraft vortex wake (vortex pair) and the ground during the takeoff and landing phases are presented. The calculations were performed within the framework of the time-dependent two-dimensional Reynolds-averaged Navier-Stokes equations using the Spalart-Shur turbulence model generalizing the turbulent viscosity transport model of Spalart and Allmaras to the case of the flows with curved streamlines and rotation. Similar calculations were carried out on the basis of the original Spalart-Allmaras model and the k- model of Menter. Some new qualitative and quantitative data on the distinctive features of the phenomenon under consideration are obtained.     

Diffusionally assisted grain-boundary sliding and viscoelasticity of polycrystals

Motivated by recent attenuation experiments on finely grained samples, we reanalyse the Raj-Ashby model of grain-boundary sliding. Two linearly elastic layers having finite thickness and identical elastic constants are separated by an interface (grain boundary) whose location is a given periodic function of position. Dissipation is confined to that interfacial region. It is caused by two mechanisms: a slip (boundary sliding) viscosity, and grain-boundary diffusion, with corresponding Maxwell relaxation times t_v and t_d. Owing to the assumption of a given, time-independent interface, the resulting boundary-value problem (b.v.p.) is linear and time-separable. The response to time-periodic forcing depends on angular frequency ω, on the ratio M=t_v/t_d of Maxwell times, and on the characteristic interface slope. The b.v.p. is solved using a perturbation method valid for small slopes. To relate features of the mechanical loss spectrum previously studied in isolation, we first discuss the solution as a function of M. Motivated by experiments, we then emphasize the case M?1 in which the relaxation times are widely separated. The loss spectrum then always has two major features: a frequency band 1?ωt_d?M^-1 within which the loss varies relatively weakly with ω; and a loss maximum at ωt_d∼M^-1 due to the slip viscosity. If corners on the interface are sufficiently rounded, those two universal features are separated by a third feature: between them, there is a strong minimum whose location is (entirely) independent of slip viscosity. The existence of that minimum has not previously been reported. These features are likely to occur even in solutions for finite interface slopes, because they are a consequence of the separation of timescales. The precise form of the spectrum in the weakly varying band must, however, be slope-dependent because it is controlled by stress singularities occurring at corners, and the strength of those singularities depends on the angle subtended by the corner.     

Development of a Low-Reynolds-number k-ω Model for FENE-P Fluids

A low-Reynolds-number k-ω model for Newtonian fluids has been developed to predict drag reduction of viscoelastic fluids described by the FENE-P model. The model is an extension to viscoelastic fluids of the model for Newtonian fluids developed by Bredberg et al. (Int J Heat Fluid Flow 23:731–743, 2002). The performance of the model was assessed using results from direct numerical simulations for fully developed turbulent channel flow of FENE-P fluids. It should only be used for drag reductions of up to 50 % (low and intermediate drag reductions), because of the limiting assumption of turbulence isotropy leading to an under-prediction of k, but compares favourably with results from k-ε models in the literature based on turbulence isotropy.     

A Bypass Transition Model Based on the Intermittency Function

An intermittency model that is formulated in local variables is proposed for representing bypass transition in Reynolds-Averaged Navier-Stokes (RANS) computations. No external data correlation is used to fix transition. Transition is initiated by diffusion, and a source term carries it to completion. A sink term is created to predict the laminar region before transition, then it vanishes in the turbulent region. Both the source and sink are functions of a wall-distance Reynolds number and turbulence scale. A modification is introduced to predict transition in separated boundary layers. The transition model is incorporated with the k?ω RANS model. The present model is implemented into a general purpose, computational fluid dynamics (CFD) code. The model is validated with several test cases. Decent agreement with the available data is observed in a range of flows.     

Numerical simulations of non‐equilibrium turbulent boundary layer flowing over a bump

Large‐eddy simulation (LES) and Reynolds‐averaged Navier–Stokes simulation (RANS) with different turbulence models (including the standard k?ε, the standard k?ω, the shear stress transport k?ω (SST k?ω), and Spalart–Allmaras (S–A) turbulence models) have been employed to compute the turbulent flow of a two‐dimensional turbulent boundary layer over an unswept bump. The predictions of the simulations were compared with available experimental measurements in the literature. The comparisons of the LES and the SST k?ω model including the mean flow and turbulence stresses are in satisfied agreements with the available measurements. Although the flow experiences a strong adverse pressure gradient along the rear surface, the boundary layer is unique in that intermittent detachment occurring near the wall. The numerical results indicate that the boundary layer is not followed by mean‐flow separation or incipient separation as shown from the numerical results. The resolved turbulent shear stress is in a reasonable agreement with the experimental data, though the computational result of LES shows that its peak is overpredicted near the trailing edge of the bump, while the other used turbulence models, except the standard k?ε, underpredicts it. Analysis of the numerical results from LES confirms the experimental data, in which the existence of internal layers over the bump surface upstream of the summit and along the downstream flat plate. It also demonstrates that the quasi‐step increase in skin friction is due to perturbations in pressure gradient. The surface curvature enhances the near‐wall shear production of turbulent stresses, and is responsible for the formation of the internal layers. The aim of the present work is to examine the response and prediction capability of LES with the dynamic eddy viscosity model as a sub‐grid scale to the complex turbulence structure with the presence of streamline curvature generated by a bumpy surface. Aiming to reduce the computational costs with focus on the mean behavior of the non‐equilibrium turbulent boundary layer of flow over the bump surface, the present investigation also explains the best capability of one of the used RANS turbulence models to capture the driving mechanism for the surprisingly rapid return to equilibrium over the trailing flat plate found in the measurements. Copyright © 2010 John Wiley & Sons, Ltd.     

Control of the aerodynamic characteristics of a profile oscillating with respect to the incidence angle in subsonic viscous flow

The results of a numerical simulation of the unsteady subsonic viscous gas flow around a two-dimensional profile oscillating with respect to the incidence angle are presented and the possibility of controlling the nonstationary aerodynamic characteristics is considered. The hysteresis phenomena typical of oscillatory profile motions are investigated, the dependence of the lift force and drag is found for various laws of periodic variation of the incidence angle with time, and the effect of the frequency and amplitude of the angular profile oscillations on the shape of the hysteresis curves is studied. The calculations were based on the numerical solution of the nonstationary Navier-Stokes equations averaged in the Reynolds sense (Reynolds equations) which were closed using the k-ω turbulence model with modeling of the laminar/turbulent transition.     

Experimental and numerical study of turbulent flow and heat transfer inside hexagonal duct

Flow and heat transfer characteristics in transition and turbulent regions are studied experimentally and numerically in a horizontal smooth regular hexagonal duct under constant wall temperature boundary condition covering a range of Reynolds number from 2.3 × 10³ to 52 × 10³. Two types of k-omega (standard and shear stress transport (SST)) and three types of k-ε (standard, renormalization (RNG), and realizable) turbulence model are employed for transition and turbulent regions, respectively. Both average and fully developed Darcy friction factor and Nusselt number are presented as a function of Reynolds number. It is seen that k-omega SST and k-ε realizable turbulence models gave the best agreement with the experimental data in transition and turbulent regions, respectively. All the experimental results are correlated within an accuracy of ±13 % and ±7 % for Nusselt number and Darcy friction factor, respectively. Results obtained in this study are compared with circular duct results using hydraulic diameter.     

An improved anisotropy-resolving subgrid-scale model for flows in laminar–turbulent transition region

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively improve the accuracy of large eddy simulation (LES) in predicting wall turbulence. Abe (2013) has recently proposed a stabilized mixed model that maintains its computational stability through a unique procedure that prevents the energy transfer between the grid-scale (GS) and SGS components induced by the scale-similarity term. At the same time, since this model can successfully predict the anisotropy of the SGS stress, the predictive performance, particularly at coarse grid resolutions, is remarkably improved in comparison with other mixed models. However, since the stabilized anisotropy-resolving SGS model includes a transport equation of the SGS turbulence energy, k_SGS, containing a production term proportional to the square root of k_SGS, its applicability to flows with both laminar and turbulent regions is not so high. This is because such a production term causes k_SGS to self-reproduce. Consequently, the laminar–turbulent transition region predicted by this model depends on the inflow or initial condition of k_SGS. To resolve these issues, in the present study, the mixed-timescale (MTS) SGS model proposed by Inagaki et al. (2005) is introduced into the stabilized mixed model as the isotropic eddy-viscosity part and the production term in the k_SGS transport equation. In the MTS model, the SGS turbulence energy, k_es, estimated by filtering the instantaneous flow field is used. Since the k_es approaches zero by itself in the laminar flow region, the self-reproduction property brought about by using the conventional k_SGS transport equation model is eliminated in this modified model. Therefore, this modification is expected to enhance the applicability of the model to flows with both laminar and turbulent regions. The model performance is tested in plane channel flows with different Reynolds numbers and in a backward-facing step flow. The results demonstrate that the proposed model successfully predicts a parabolic velocity profile under laminar flow conditions and reduces the dependence on the grid resolution to the same degree as the unmodified model by Abe (2013) for turbulent flow conditions. Moreover, it is shown that the present model is effective at transitional Reynolds numbers. Furthermore, the present model successfully provides accurate results for the backward-facing step flow with various grid resolutions. Thus, the proposed model is considered to be a refined anisotropy-resolving SGS model applicable to laminar, transitional, and turbulent flows.     

Compound wall treatment with low‐Re turbulence model

A generalized treatment for the wall boundary conditions relating to turbulent flows is developed that blends the integration to a solid wall with wall functions. The blending function ensures a smooth transition between the viscous and turbulent regions. An improved low Reynolds number k?ε model is coupled with the proposed compound wall treatment to determine the turbulence field. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. Computations with fine and coarse meshes of a few flow cases yield appreciably good agreement with the direct numerical simulation and experimental data. The method is recommended for computing the complex flows where computational grids cannot satisfy a priori the prerequisites of viscous/turbulence regions. Copyright © 2011 John Wiley & Sons, Ltd.