Shock-unsteadiness model applied to oblique shock wave/turbulent boundary-layer interaction

Reynolds-averaged Navier–Stokes prediction of shock wave/turbulent boundary layer interactions can yield significant error in terms of the size of the separation bubble. In many applications, this can alter the shock structure and the resulting surface properties. Shock-unsteadiness modification of Sinha et al. (Physics of Fluids, Vol.15, No.8, 2003) has shown potential in improving separation bubble prediction in compression corner flows. In this article, the modification is applied to oblique shock wave interacting with a turbulent boundary layer. The challenges involved in the implementation of the shock-unsteadiness correction in the presence of multiple shock waves and expansion fans are addressed in detail. The results show that a robust implementation of the model yields appreciable improvement over standard k–ω turbulence model predictions.     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

Acceleration on stretched meshes with line-implicit LU-SGS in parallel implementation

The implicit lower–upper symmetric Gauss–Seidel (LU-SGS) solver is combined with the line-implicit technique to improve convergence on the very anisotropic grids necessary for resolving the boundary layers. The computational fluid dynamics code used is Edge, a Navier–Stokes flow solver for unstructured grids based on a dual grid and edge-based formulation. Multigrid acceleration is applied with the intention to accelerate the convergence to steady state. LU-SGS works in parallel and gives better linear scaling with respect to the number of processors, than the explicit scheme. The ordering techniques investigated have shown that node numbering does influence the convergence and that the orderings from Delaunay and advancing front generation were among the best tested. 2D Reynolds-averaged Navier–Stokes computations have clearly shown the strong efficiency of our novel approach line-implicit LU-SGS which is four times faster than implicit LU-SGS and line-implicit Runge–Kutta. Implicit LU-SGS for Euler and line-implicit LU-SGS for Reynolds-averaged Navier–Stokes are at least twice faster than explicit and line-implicit Runge–Kutta, respectively, for 2D and 3D cases. For 3D Reynolds-averaged Navier–Stokes, multigrid did not accelerate the convergence and therefore may not be needed.     

An inviscid/viscous coupling approach for vortex flow field calculations

A new computational approach is developed for the analysis of vortex-dominated flow fields around highly swept wings at high angles of attack. In this approach an inviscid Euler technology is coupled with viscous models, similar to inviscid/boundary layer coupling. The viscous nature of the vortex core is represented by an algebraic model derived from the Navier–Stokes equations. The approach also accounts for the effects of the viscous shear layer near a wing surface through a modified surface boundary condition. The inviscid/viscous coupling consistently provides improved predictions of leading edge separation, vortex bursting and secondary vortex formation at relatively low computational cost. Results for several cases are compared with wind tunnel tests and other Euler and Navier-Stokes solutions.     

Numerical modeling and experimental measurements of water spray impact and transport over a cylinder

This study compares experimental measurements and numerical simulations of liquid droplets over heated (to a near surface temperature of 423 K) and unheated cylinders. The numerical model is based on an unsteady Reynolds-averaged Navier–Stokes (RANS) formulation using a stochastic separated flow (SSF) approach for the droplets that includes submodels for droplet dispersion, heat and mass transfer, and impact on a solid surface. The details of the droplet impact model are presented and the model is used to simulate water spray impingement on a cylinder. Computational results are compared with experimental measurements using phase Doppler interferometry (PDI). Overall, good agreement is observed between predictions and experimental measurements of droplet mean size and velocity downstream of the cylinder.     

An efficient upwind/relaxation algorithm for the Euler and Navier–Stokes equations

An efficient Euler and full Navier–Stokes solver based on a flux splitting scheme is presented. The original Van Leer flux vector splitting form is extended to arbitrary body-fitted co-ordinates in the physical domain so that it can be used with a finite volume scheme. The block matrix is inverted by Gauss–Seidel iteration. It is verified that the often used reflection boundary condition will produce incorrect flux crossing the wall and cause too large numerical dissipation if flux vector splitting is used. To remove such errors, an appropriate treatment of wall boundary conditions is suggested. Inviscid and viscous steady transonic internal flows are analysed, including the case of shock-induced boundary layer separation.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical study of gas-cyclone airflow: an investigation of turbulence modelling approaches

A numerical study of unsteady single-phase vortical flow inside a cyclone is presented. Two different geometric configurations have been considered, with the goal of assessing several different turbulence modelling approaches for this class of problem. The models investigated include three Reynolds-averaged Navier–Stokes models: a commonly used two-equation eddy-viscosity model, a differential Reynolds stress model (DRSM) and an eddy-viscosity model sensitised to rotational and curvature (RC) effects which was recently developed and implemented into a commercial CFD (computational fluid dynamics) code by the authors. Results were also obtained using large eddy simulation (LES). The computational results are analysed and compared with available experimental data. The RC-sensitised eddy-viscosity model shows significant improvement over the standard eddy-viscosity model. The RC-sensitised model, DRSM and LES model predictions of the mean flowfield are in good agreement with the experimental data. The results suggest that curvature- and rotation-sensitive eddy-viscosity models may provide a practical alternative to more computationally intensive approaches.     

Unsteady free surface wave-induced separation: Vortical structures and instabilities

Vortical structures and instability mechanisms of the unsteady free surface wave-induced separation around a surface-piercing NACA0024 foil at a Froude number of 0.37 and a Reynolds number of 1.52×10⁶ are studied using an unsteady Reynolds-averaged Navier–Stokes (URANS) code with a blended k?ε/k?ω turbulence model and a free surface tracking method. At the free surface, the separated flow reattaches to the foil surface resulting in a wall-bounded separation bubble. The mean and instantaneous flow topologies in the separation region are similar to the owl-face pattern. The initial shear-layer instability, the Karman-like instability, and the flapping instability are identified, and their scaling and physical mechanisms are studied. Validation with experimental fluid dynamics (EFD) and comparison with complementary detached-eddy simulation (DES) indicate that URANS resolves part of the organized oscillations due to the large-scale unsteady vortical structures and instabilities, thereby capturing the gross features of the unsteady separation. The URANS solutions show an initial amplitude defect of 30% for the free surface oscillations where the shear layer separates, and the defect progressively increases downstream as URANS rapidly dissipates the rolled up vortices.     

Buffeting in transonic flow prediction using time‐dependent turbulence model

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and flow separations on wing upper surface induce flow instabilities, ‘buffet’, and then the buffeting (structure vibrations). This phenomenon can greatly influence the aerodynamic performance. These flow excitations are self‐sustained and lead to a surface effort due to pressure fluctuations. They can produce enough energy to excite the structure. The objective of the present work is to predict this unsteady phenomenon correctly by using unsteady Navier–Stokes‐averaged equations with a time‐dependent turbulence model based on the suitable (k–ε) turbulent eddy viscosity model. The model used is based on the turbulent viscosity concept where the turbulent viscosity coefficient (Cμ) is related to local deformation and rotation rates. To validate this model, flow over a flat plate at Mach number of 0.6 is first computed, then the flow around a NACA0012 airfoil. The comparison with the analytical and experimental results shows a good agreement. The ONERA OAT15A transonic airfoil was chosen to describe buffeting phenomena. Numerical simulations are done by using a Navier–Stokes SUPG (streamline upwind Petrov–Galerkin) finite‐element solver. Computational results show the ability of the present model to predict physical phenomena of the flow oscillations. The unsteady shock wave/boundary layer interaction is described. Copyright © 2005 John Wiley & Sons, Ltd.     

Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems

A numerical solution for shallow-water flow is developed based on the unsteady Reynolds-averaged Navier–Stokes equations without the conventional assumption of hydrostatic pressure. Instead, the non-hydrostatic pressure component may be added in regions where its influence is significant, notably where bed slope is not small and separation in a vertical plane may occur or where the free-surface slope is not small. The equations are solved in the σ-co-ordinate system with semi-implicit time stepping and the eddy viscosity is calculated using the standard k–ϵ turbulence model. Conventionally, boundary conditions at the bed for shallow-water models only include vertical diffusion terms using wall functions, but here they are extended to include horizontal diffusion terms which can be significant when bed slope is not small. This is consistent with the inclusion of non-hydrostatic pressure. The model is applied to the 2D vertical plane flow of a current over a trench for which experimental data and other numerical results are available for comparison. Computations with and without non-hydrostatic pressure are compared for the same trench and for trenches with smaller side slopes, to test the range of validity of the conventional hydrostatic pressure assumption. The model is then applied to flow over a 2D mound and again the slope of the mound is reduced to assess the validity of the hydrostatic pressure assumption. © 1998 John Wiley & Sons, Ltd.     

An advanced switching parameter for a hybrid LES/RANS model considering the characteristics of near-wall turbulent length scales

In this study, we proposed an idea for an advanced switching parameter used in a hybrid approach connecting large eddy simulation (LES) with Reynolds-averaged Navier–Stokes modeling [the hybrid LES/RANS (HLR) model]. Although the HLR model is promising way to predict engineering turbulent flows, an important problem is that RANS is always adopted in the near-wall region, even if the grid resolution is fine enough for LES. To overcome this difficulty, the switching parameter proposed here introduced knowledge of the Kolmogorov microscale that is thought to be reasonable for representing the near-wall turbulence. This parameter enabled the present HLR model to be smoothly replaced by a full LES if a grid resolution was fine enough in the near-wall region. To confirm model performance, the present HLR model was applied to numerical simulations of a periodic hill flow as well as fundamental plane channel flows. The model generally provided reasonable predictions for these test cases that include complex turbulence with massive flow separation.     

Euler–Euler large eddy simulations for dispersed turbulent bubbly flows

In this paper we present detailed Euler–Euler Large Eddy Simulations (LES) of dispersed bubbly flow in a rectangular bubble column. The motivation of this study is to investigate the potential of this approach for the prediction of bubbly flows, in terms of mean quantities. The physical models describing the momentum exchange between the phases including drag, lift and wall force were chosen according to previous experiences of the authors. Experimental data, Euler–Lagrange LES and unsteady Euler–Euler Reynolds-Averaged Navier–Stokes results are used for comparison. It is found that the present model combination provides good agreement with experimental data for the mean flow and liquid velocity fluctuations. The energy spectrum obtained from the resolved velocity of the Euler–Euler LES is presented as well.     

An improved boundary element method for modelling a self-reacting point absorber wave energy converter

A numerical model based on a boundary element method (BEM) is developed to predict the performance of two-body self-reacting floating-point absorber (SRFPA) wave energy systems that operate predominantly in heave. The key numerical issues in applying the BEM are systematically discussed. In particular, some improvements and simplifications in the numerical scheme are developed to evaluate the free surface Green’s function, which is a main element of difficulty in the BEM. For a locked SRFPA system, the present method is compared with the existing experiment and the Reynolds-averaged Navier–Stokes (RANS)-based method, where it is shown that the inviscid assumption leads to substantial over-prediction of the heave response. For the unlocked SRFPA model we study in this paper, the additional viscous damping primarily induced by flow separation and vortex shedding, is modelled as a quadratic drag force, which is proportional to the square of body velocity. The inclusion of viscous drag in present method significantly improves the prediction of the heave responses and the power absorption performance of the SRFPA system, obtaining results excellent agreement with experimental data and the RANS simulation results over a broad range of incident wave periods, except near resonance in larger wave height scenarios. It is found that the wave overtopping and the re-entering impact of out-of-water floating body are observed more frequently in larger waves, where these non-linear effects are the dominant damping sources and could significantly reduce the power output and the motion responses of the SRFPA system.     

On wall pressure fluctuations and their coupling with vortex dynamics in a separated–reattached turbulent flow over a blunt flat plate

This study deals with the numerical predictions through Large-Eddy Simulation (LES) of the separated–reattached turbulent flow over a blunt flat plate for analyzing main coherent structure features and their relation to the unsteady pressure field. A compressible approach that inherently includes acoustic propagation is here followed to describe the relationship between pressure fluctuations and vortex dynamics around the separation bubble. The objective of the present work is then to contribute to a better understanding of the coupling between the vortex dynamics and the wall pressure fluctuations. The filtered compressible Navier–Stokes equations are then solved with a numerical method that follows a Lax–Wendroff approach to recover a high accuracy in both time and space. For validations, the present numerical results are compared to experimental measurements, coming from both the Pprime laboratory (Sicot el al., 2012) and the literature (Cherry et al., 1984; Kiya and Sasaki, 1985; Tafti and Vanka,1991; Sicot et al., 2012). Our numerical results very well predict mean and fluctuating pressure and velocity fields. Flapping, shedding as well as Kelvin–Helmholtz characteristic frequencies educed by present simulations are in very good agreement with the experimental values generally admitted. These characteristic modes are also visible on unsteady pressure signatures even far away from the separation. Spectral, POD and EPOD (extended POD) analyses are then applied to these numerical data to enhance the salient features of the pressure and velocity fields, especially the unsteady wall pressure in connection with either the vortex shedding or the low frequency shear-layer flapping. A contribution to the understanding of the coupling between wall pressure fluctuations and eddy vortices is finally proposed.     

Numerical study of turbulent trailing-edge flows with base cavity effects using URANS

Turbulent flows over lifting surfaces exhibiting trailing-edge vortex shedding often cause adverse and complex phenomena, such as self-induced vibration and noise. In this paper, a numerical study on flow past a blunt-edged two-dimensional NACA 0015 section and the same section with various base cavity shapes and sizes at high Reynolds numbers has been performed using the unsteady Reynolds-averaged Navier–Stokes (URANS) approach with the realisable κ–ε turbulence model. The equations are solved using the control volume method of second-order accuracy in both spatial and time domains. The assessment of the application of URANS for periodic trailing-edge flow has shown that reasonable agreement is achieved for both the time-averaged and fluctuating parameters of interest, although some differences exist in the prediction of the near-wake streamwise velocity fluctuation magnitudes. The predicted Strouhal numbers of flows past the squared-off blunt configuration with varying degrees of bluntness agree well with published experimental measurements. It is found that the intensity of the vortex strengths at the trailing-edge is amplified when the degree of bluntness is increased, leading to an increase in the mean square pressure fluctuations. The numerical prediction shows that the presence of the base cavity at the trailing-edge does not change the inherent Strouhal number of the 2D section examined. However, it does have an apparent effect on the wake structure, local pressure fluctuations and the lift force fluctuations. It is observed that the size of the cavity has more influence on the periodic trailing-edge flow than its shape does.     

Finite element modelling of local scour below a pipeline under steady currents

Local scour has been identified as the main factor that causes failures of structures in offshore engineering. Numerous research efforts have been devoted to local scour around offshore pipelines in the past. In this paper, a finite element numerical model is established for simulating local scour below offshore pipelines in steady currents. The flow is simulated by solving the unsteady Reynolds-averaged Navier–Stokes equations with a standard k ? ? turbulent model closure. A sand slide scheme is proposed for the scour calculation, and bed load is considered in the proposed scour model. To account for changes in bed level, the moving mesh method is adopted to capture the water–sediment interface (bed), and the change of bed level is calculated by solving Exner–Polya equation. All the equations are discretised within the two-step Taylor–Galerkin algorithm in this paper. It is found that the sand slide model works well for the simulation of the scour, and the numerical results are shown to be in good agreement with the available experimental data.     

Interactive boundary layer models for channel flow

In this paper, the aim is to present the results of a new approach for the asymptotic modeling of two-dimensional steady, incompressible, laminar flows in a channel. More precisely, for high Reynolds numbers, the walls of the channel are deformed in such a way that separation is possible. Of course, numerical solutions of Navier–Stokes equations can be calculated but it is believed that an asymptotic analysis helps in the understanding of the flow structure. Numerical solutions of Navier–Stokes equations are compared with solutions of asymptotic models included in a more general model called global interactive boundary layer.     

On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid

For incompressible Navier–Stokes equations in primitive variables, a method of setting absorbing outflow boundary conditions on an artificial boundary is considered. The advection equations used on the outflow boundary are convenient for finite difference (FD) methods, where a weak formulation of a problem is inapplicable. An unsteady viscous incompressible Navier–Stokes flow in a channel with a moving damper is modeled. An accurate comparison and analysis of numerical and mechanical situations are carried out for a variety of boundary conditions and Reynolds numbers. The proposed outflow conditions provide that the problem with Dirichlet boundary conditions should be solved on each time step.     

NUMERICAL CALCULATION OF 2D,UNSTEADY FLOW IN CENTRIFUGAL PUMPS: IMPELLER AND VOLUTE INTERACTION

A numerical model is developed for calculating the two-dimensional, unsteady, incompressible and turbulent flow within the rotating impeller and stationary volute of an industrial centrifugal pump. The objective is the investigation and comprehension of the instantaneous behaviour of centrifugal pumps, aiming at the reduction of vibrations, radial forces and hydraulic noise. The computation is performed within a blade-to-blade streamtube for the impeller and a tube normal to the axis of rotation for the volute. The equations to be solved are the unsteady Reynolds-averaged Navier–Stokes equations along with the continuity equation and the unsteady κ–ϵ equations for turbulence modelling. The finite volume method is applied for space discretization and an implicit scheme for time discretization. A multidomain overlapping grid technique is used for matching together the relative flow field calculated within the rotating impeller and the absolute one calculated within the stationary volute. In this way the impeller and volute interaction is directly taken into account. The numerical model is validated for a centrifugal pump of N _q=32 under design flow conditions. Comparisons between calculation and measurements show fairly good agreement.