Erik Dick

MODELLING OF BYPASS TRANSITION WITH CONDITIONED NAVIER–STOKES EQUATIONS COUPLED TO AN INTERMITTENCY TRANSPORT EQUATION

A differential method is proposed to simulate bypass transition. The intermittency in the transition zone is taken into account by conditioned averages. These are averages taken during the fraction of time the flow is turbulent or laminar respectively. Starting from the Navier-Stokes equations, conditioned continuity, momentum and energy equations are derived for the larninar and turbulent parts of the intermittent flow. The turbulence is described by a classical k-e model. The supplementary parameter, the intermittency factor, is determined by a transport equation applicable for zero, favourable and adverse pressure gradients. Results for these pressure gradients are given.

Computer Simulation of Intraventricular Flow and Pressure Gradients During Diastole

A two-dimensional axisymmetric computer model is developed for the simulation of the filling flow in the left ventricle (LV). The computed results show that vortices are formed during the acceleration phases of the filling waves. During the deceleration phases these are amplified and convected into the ventricle. The ratio of the maximal blood velocity at the mitral valve (peak E velocity) to the flow wave propagation velocity (WPV) of the filling wave is larger than 1. This hemodynamic behavior is also observed in experiments in vitro (Steen and Steen, 1994, Cardiovasc. Res., 28, pp. 1821-1827) and in measurements in vivo with color M-mode Doppler echocardiography (Stugaard et al., 1994, J. Am. Coll. Cardiol., 24, 663-670). Computed intraventricular pressure profiles are similar to observed profiles in a dog heart (Courtois et al., 1988, Circulation, 78, pp. 661-671). The long-term goal of the computer model is to study the predictive value of noninvasive parameters (e.g., velocities measured with Doppler echocardiography) on invasive parameters (e.g., pressures, stiffness of cardiac wall, time constant of relaxation). Here, we show that higher LV stiffness results in a smaller WPV for a given peak E velocity. This result may indicate an inverse relationship between WPV and LV stiffness, suggesting that WPV may be an important noninvasive index to assess LV diastolic stiffness, LV diastolic pressure and thus atrial pressure (preload).

Modeling of laminar-turbulent transition for high freestream turbulence

To simulate transitional skin friction or heat transfer, the conditionally averaged Navier-Stokes equations are used. To describe the diffusion of freestream turbulence into the boundary layer and the intermittent laminar-turbulent flow behavior during transition, a turbulence weighting factor τ is used. A transport equation is presented for this τ-factor including convection, diffusion, production, and sink terms, In combination with the conditioned Navier-Stokes equations, this leads to an accurate calculation of flow characteristics within the transitional layer. The method is validated on transitional skin friction and heat transfer measurements, respectively on a flat plate and in a linear turbine cascade

Heat transfer predictions with a cubic k–ε model for axisymmetric turbulent jets impinging onto a flat plate

Abstract Local heat transfer in turbulent axisymmetric jets, impinging onto a flat plate, is predicted with a cubic k – e model. Both the constitutive law for the Reynolds stresses and the transport equation for the dissipation rate e contribute to improved heat transfer predictions. The stagnation point value and the shape of the profiles of the Nusselt number are well predicted for different distances between the nozzle and the flat plate. Accurate flow field predictions, obtained with the presented turbulence model, are the basis for the quality of the heat transfer results. The influence of the nozzle–plate distance on the stagnation point Nusselt number, is also correctly captured. For a fixed nozzle–plate distance, the influence of the Reynolds number on the stagnation point heat transfer is correctly reproduced. Comparisons are made to experimental data and to results from a low-Reynolds standard k – e model [1] and the v 2 – f model [2] .

One-equation RG hybrid RANS/LES computation of a turbulent impinging jet

A one-equation variant of a previously developed two-equation, renormalization group based, hybrid RANS/LES model is presented. The model consists of a transport equation for the mean dissipation rate and an algebraically prescribed length scale. The length scale is proportional to the wall-distance in RANS regions of the flow and to the filter width in LES regions. A very simple, but efficient, near-wall model is also presented, in the manner of Yakhot and Orszags RNG modeling. The only free parameter entering the low-Reynolds formulation has been calibrated against channel flow, and the value corresponds well with experimental values for the same parameter obtained for homogeneous isotropic turbulence. The model is then validated for a turbulent impinging jet heat transfer problem. Results are satisfactory and compare very favorably against DES and dynamic LES for the same computation.

Computational Treatment of Source Terms in Two-Equation Turbulence Models

The source terms in turbulence models require careful treatment to obtain a stable discretization. The choice between implicit and explicit treatment has to be made. This can be done either on the basis of individual terms or on the basis of the exact Jacobian of the source terms. A comparison of both methods shows that the latter is generally applicable and superior to the first, approximate method with respect to convergence speed. This comes from the possibility of using the multigrid technique with the exact method, whereas this is not always possible with the approximate method. In principle, for robustness a time-step restriction for the source terms has to be introduced to prevent the turbulence quantities from becoming negative or infinitely large. An approximation of the appropriate time step is calculated. Practical results, however, indicate that the time-step restriction is not always necessary. Different two-equation turbulence models are investigated confirming the generality of the approach

Application of a New Cubic Turbulence Model to Piloted and Bluff-Body Diffusion Flames

A new two-equation turbulence model is described. It combines an algebraic, non-linear expression of the Reynolds stresses in terms of strain rate and vorticity tensor components, with a modified transport equation for the dissipation rate. Thanks to the cubic law for the Reynolds stresses, the influence on turbulence from streamline curvature is accounted for, while the increase in computational costs is small. The classical transport equation for the dissipation rate is altered, in order to bring more physics into this equation. As a result, more realistic values for the turbulence quantities are obtained. A new low-Reynolds source term has been introduced and a model parameter is written in terms of dimensionless strain rate and vorticity. The resulting model is firstly applied to the inert turbulent flow over a backward-facing step, demonstrating the quality of the turbulence model. Next, application to an inertly mixing round jet reveals that the spreading rate of the mixture fraction is correctly predicted. Afterwards, a piloted-jet diffusion flame is considered. Finally, inert and reacting flows in a bluff-body burner are addressed. It is illustrated for both reacting test cases that the turbulence model is important with respect to the flame structure. It is more important than the chemistry model for the chosen test cases. Results are compared to what is obtained by linear turbulence models. For the reacting test cases, the conserved scalar approach with pre-assumed β-probability density function (PDF) is used.

Analysis and Stabilization of Fluid-Structure Interaction Algorithm for Rigid-Body Motion

Fluid-structure interaction computations in geometries where different chambers are almost completely separated from each other by a movable rigid body but connected through very small gaps can encounter stability problems when a standard explicit coupling procedure is used for the coupling of the fluid flow and the movement of the rigid body. An example of such kind of flows is the opening and closing of valves, when the valve motion is driven by the flow. A stability analysis is performed for the coupling procedure of the movement of a cylinder in a cylindrical tube, filled with fluid, Between the moving cylinder and the tube, a small gap is present, so that two chambers are formed. It is shown that a standard explicit coupling procedure or an implicit coupling procedure with explicit coupling in the subiterations steps can lead to unstable motion depending on the size of the gaps, the density of the rigid body, and the density of the fluid. It is proven that a reduction of the time-step size cannot stabilize the coupling procedure. An implicit coupling procedure with implicit coupling in the subiterations has to be used. An illustration is given on how such a coupling procedure can be implemented in a commercial computational fluid dynamics (CFD) software package. The CFD package FLUENT (Fluent, Inc.) is used. As an application, the opening and the closing of a prosthetic aortic valve is computed.

On the performance of relaxation filtering for large-eddy simulation

In this work, the performance of large-eddy simulation (LES) based on the relaxation-filtering (RF) technique has been investigated quantitatively. In RF-based LES, the velocity field is filtered each nth time step, using a standard finite-difference filter, characterized by a specific order of accuracy m, and a fixed filtering strength σ. Hence, the procedure dissipates the amount of energy related to the residual stresses, and thus models the dissipative effect of the unresolved scales on the resolved scales. Since the order m and strength σ are related to the spectral distribution and the magnitude of the dissipation, respectively, these predefined parameters are crucial for the success of the method. Here, their influence is systematically investigated for the Taylor–Green vortex flow at a Reynolds number of 3000. First, the effects of m and σ are studied a priori in Fourier space. Further, 36 LESs are performed, each with a different combination of order m=4, 6, 8, 10, 12, 14 and strength σ=0.15, 0.2, 0.4, 0.6, 0.8, 1, and the turbulent statistics are compared with those of a direct numerical simulation, filtered at identical resolutions. The a priori, as well as the a posteriori results indicate that, for low filter orders m⩽4, the LES accuracy is rather poor and depends strongly on the filtering strength σ. However, for higher order filters, i.e. m⩾8, the accuracy is quite good and the results, including the resolved and subgrid dissipation rates, are nearly independent of the strength σ for σ⩾0.4. In this case, the spectral dissipation-distribution, determined by m, turns out to be the dominant parameter, whereas the dissipation strength, determined by σ, is of minor importance.

Numerical study of natural convective heat transfer with large temperature differences.

Steady‐state two‐dimensional solutions to the full compressible Navier‐Stokes equations are computed for laminar convective motion of a gas in a square cavity with large horizontal temperature differences. No Boussinesq or low‐Mach number approximations of the Navier‐Stokes equations are used. Results for air are presented. The ideal‐gas law is used and viscosity is given by Sutherland’s law. An accurate low‐Mach number solver is developed. Here an explicit third‐order discretization for the convective part and a line‐implicit central discretization for the acoustic part and for the diffusive part are used. The semi‐implicit line method is formulated in multistage form. Multigrid is used as the acceleration technique. Owing to the implicit treatment of the acoustic and the diffusive terms, the stiffness otherwise caused by high aspect ratio cells is removed. Low Mach number stiffness is treated by a preconditioning technique. By a combination of the preconditioning technique, the semi‐implicit discretization and the multigrid formulation a convergence behaviour is obtained which is independent of grid size, grid aspect ratio, Mach number and Rayleigh number. Grid converged results are shown for a variety of Rayleigh numbers.