An Oseen-type model for swirling internal separated flows

The viscous, laminar, separated flow downstream of a sudden expansion in a pipe is studied. The flow is modeled by an Oseen-type equation, but with the additional feature that the nonlinearity in the swirl is retained. Exact solutions are obtained for a high-Reynolds-number limit and for arbitrary Reynolds number by use of an eigenfunction-expansion procedure, in the presence of swirl. This leads to a non-standard eigenvalue problem. When the swirl is sufficiently large, a central recirculating region is observed. The effect of the pressure gradients on the velocity profiles and the central recirculating eddy is discussed. The low-Reynolds-number solutions go over smoothly to the large Reynolds number solution as the Reynolds number increases. Good agreement is obtained with the numerically computed value of the reattachment length.     

Concerning inviscid solutions for large-scale separated flows

Summary The large-scale separated eddies set up behind a bluff body at high Reynolds number are considered, for steady laminar planar flow. The main eddies are massive and are controlled predominantly by inviscid mechanics, with uniform vorticity inside. Analytical and computational solutions of the massive-eddy (vortex-sheet) problem are then described. A further possibility studied is that, even with lateral symmetry assumed, there may still be an extra degree of nonsymmetry of skewing with respect to the streamwise direction. Small-scale separations, where a Benjamin-Ono equation also possibly yielding nonsymmetric solutions can come into play, are discussed briefly.     

Inviscid separated flows of finite extent

Two-dimensional, inviscid, incompressible flow is considered when the flow region contains a separation bubble of finite length. Within the separation bubble a slender-eddy approximation is employed, whilst outside it small disturbance theory is used to solve the potential-flow equations. The solution is completed by matching the pressure across the vortex sheet that divides the two regions of flow. Solutions are presented for the flow past smooth indentations in an otherwise plane boundary.     

On body-fitted co-ordinates for separated laminar flows past wall-mounted obstacles

A body-fitted curvilinear co-ordinate system is used to solve the equations of two-dimensional incompressible laminar flow over bluff obstructions by finite differences. Arbitrary conditions at the corner are removed by this method. Results for a backward-facing step are in reasonable agreement with those obtained with conventional mesh systems, and the differences are explained. A treatment of a channel expansion, in comparison with empirical data, is also included. The capability of the present method to handle arbitrary two-dimensional geometries is stressed and demonstrated, using a triangle and a semi-circle as examples.     

A finite difference calculation procedure for three-dimensional turbulent separated flows

A general finite difference scheme has been proposed along with a three-dimensional co-ordinate transformation procedure for the prediction of three-dimensional fully elliptic flows. This numerical scheme has been successfully employed for the calculations of the three-dimensional turbulent separated flow in a rectangular diffuser. The complexity of the phenomena is seen to increase tremendously for the three-dimensional flows of this class.     

Numerical simulation of gas-particle flows behind a backward-facing step using an improved stochastic separated flow model

An improved stochastic separated flow (ISSF) model developed by the present authors is tested in gas-particle flows behind a backward-facing step, in this paper. The gas phase of air and the particle phase of 150 μm glass and 70 μm copper spheres are numerically simulated using the k–ɛ model and the ISSF model, respectively. The predicted mean streamwise velocities as well as streamwise and transverse fluctuating velocities of both phases agree well with experimental data reported by Fessler. The reattachment length of 7.6H matches well with the experimental value of 7.4H. Distributions of particle number density are also given and found to be in good agreement with the experiment. The sensitivity of the predicted results to the number of calculation particles is studied and the improved model is shown to require much less calculation particles and less computing time for obtaining reasonable results as compared with the traditional stochastic separated flow model. It is concluded that the ISSF model can be used successfully in the prediction of backward-facing step gas-particle flows, which is characterised by having recirculating regions and anisotropic fluctuating velocities. Received 20 June 2000     

Experimental study of internal cavitating flows

The parameters of a cavitating liquid are determined on the basis of measurements of the initial flow parameters and forces acting on the nozzle. The volume concentrations of a cavitating flow are also measured by a conductometric method simultaneously with a determination of their values from measurements of the forces acting on the nozzle.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 26, No. 2, pp. 309–313, February, 1974.     

Numerical simulation of the separated fluid flows at large Reynolds numbers

The variety of flow regimes (steady separated, periodically separated-‘Karman vortex street’, unsteady turbulent) and their characteristic peculiarities (separation and reattachment points, secondary separation, boundary layer, instability of the shear mixing layer, etc.) require the construction of effective numerical methods, which will be able to simulate adequately the considered flows. MERANGE ? SMIF–a splitting method for physical factors of incompressible fluids¹-is used for calculations of the steady and unsteady fluid flows past a circular cylinder in a wide range of Reynolds numbers (10° < Re < lo6). The finite-difference scheme for this method is of second order accuracy in the space variables, has minimal numerical viscosity and is also monotonic. Use of the Navier-Stokes equations with the corresponding transformation of Cartesian co-ordinates allows the calculations to be made by one algorithm both in a boundary layer and out of it. The method allows calculations at Re = ∞ cc and simulation of d‘Alembert’s paradox. Some results on the classical problem of the flow around a circular cylinder for a wide range of Reynolds numbers are discussed. The crisis of the total drag coefficient and the sharp rise of the Strouhal number are simulated numerically (without any turbulence models) for the critical Reynolds numbers (Re ≈ 4 × 10⁵), and are in a good agreement with experimental data.     

Petrov-Galerkin finite element model for compressible flows

A Petrov-Galerkin method for the solution of the compressible Euler and Navier-Stokes equations is presented. It is based on the introduction of an anisotropic blancing diffusion in the direction of the local direction of propagation of the scalar variables. The local direction in which the anisotropic diffusion is introduced is uniquely determined, and the magnitude of the balancing diffusion is automatically calculated locally using a criterion that is optimal for one-dimensional transport equations. The algorithm has been implemented using four-noded bilinear elements with forward Euler and second-order Runge-Kutta methods of integration in time. Several applications are presented and show the stability and approximation properties of the method.     

An improved stochastic separated flow model for turbulent two-phase flow

 An improved stochastic separated flow model is proposed to obtain reasonable statistical characteristics of a two-phase flow. Effects of the history of a particle and its current trajectory position on the mean-square fluctuating velocity of the dispersed phase are continuously considered in this model. Comparing with the conventional model, results using the improved model are more reasonable and can also be obtained more easily. Furthermore, the improved model requires less computational particles for simulating dispersed-phase turbulence at the beginning of the stochastic trajectory. In this paper, an application in turbulent two-phase flow of planar mixing layer is carried out. Numerical results including velocity, mean-square fluctuating velocity, particle number density and pdf of fluctuation velocity of dispersed phase are shown to compare well with experimental data.     

The use of the hybrid RANS/ILES approach for the investigation of three-dimensional separated turbulent flows in curvilinear diffusers

The hybrid RANS/ILES approach is used for the investigation of turbulent separated flow in curvilinear annular diffusers with the area ratio of 2.04 and 2.7. The effect of geometry on the loss of symmetry of flow in an axisymmetric annular diffuser is considered. The effect of pressure difference in the diffuser on the flow and on the characteristics of turbulence in this diffuser is investigated. The effect of nonuniform total pressure at the channel inlet on the distribution of parameters in the outlet section of the diffuser is determined. The loss of total pressure in diffusers is determined for all of the considered modes. The accuracy of the results is confirmed by comparison with the experimental data available for these diffusers.     

A two-phase model for dry density-varying granular flows

A granular flow is normally comprised of a mixture of grain-particles (such as sand, gravel or rocks) of different sizes. In this study, dry granular flows are modeled utilizing a set of equations akin to a two-phase mixture system, in which the interstitial fluid is air. The resultant system of equations for a two-dimensional configuration includes two continuity and two momentum balance equations for the two respective constituents. The density variation is described considering the phenomenon of air entrainment/extrusion at the flow surface, where the entrainment rate is assumed to be dependent on the divergent or convergent behavior of the solid constituent. The density difference between the two constituents is extremely large, so, as a consequence scaling analysis reveals that the flow behavior is dominated by the solid species, yielding small relative velocities between the two constituents. A non-oscillatory central (NOC) scheme with total variation diminishing (TVD) limiters is implemented. Three numerical examples are investigated: the first being related to the flow behaviors on a horizontal plane with an unstable initial condition; the second example is devoted to simulating a dam-break problem with respect to different initial conditions; and in the third one investigates the behavior of a finite mass of granular material flowing down an inclined plane. The key features and the capability of the equations to model the behavior are illustrated in these numerical examples.     

A two-fluid model for avalanche and debris flows

Geophysical mass flows--debris flows, avalanches, landslides--can contain O(10(6)-10(10)) m(3) or more of material, often a mixture of soil and rocks with a significant quantity of interstitial fluid. These flows can be tens of meters in depth and hundreds of meters in length. The range of scales and the rheology of this mixture presents significant modelling and computational challenges. This paper describes a depth-averaged 'thin layer' model of geophysical mass flows containing a mixture of solid material and fluid. The model is derived from a 'two-phase' or 'two-fluid' system of equations commonly used in engineering research. Phenomenological modelling and depth averaging combine to yield a tractable set of equations, a hyperbolic system that describes the motion of the two constituent phases. If the fluid inertia is small, a reduced model system that is easier to solve may be derived.     

Influence of heat and mass transfer on the development of two-dimensional separated flows

The effect of cooling and heating of a streamlined surface, free mass transfer, weak and strong isothermal injection, and suction on the development of supersonic turbulent separated flows is considered. The influence of the temperature factor and inefficiency ratio parameter on the gas-dynamic and geometric, characteristics of separated turbulent flows is estimated. Dnepropetrovsk State University, Dnepropetrovsk Institute of Chemical Engineering. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 69,No. 4, pp. 647–656, July–August, 1996.     

Interference of separated flows behind backward-facing step in the presence of passive control

The interaction of turbulent separated flows behind a backward-facing step in the presence of a passive miniturbulizer has been experimentally studied using digital particle-tracking velocimetry techniques. It is established that an obstacle placed in front of the step modifies the profiles of velocity and turbulent pulsation and significantly changes the length of a recirculation zone.     

Finite element analysis of internal flows with heat transfer

This paper presents a finite element-based model for the prediction of 2-D and 3-D internal flow problems. The Eulerian velocity correction method is used which can render a fast finite element code comparable with the finite difference methods. Nine different models for turbulent flows are incorporated in the code. A modified wall function approach for solving the energy equation with high Reynolds number models is presented for the first time. This is an extension of the wall function approach of Benim and Zinser and the method is insensitive to initial approximation. The performance of the nine turbulent models is evaluated by solving flow through pipes. The code is used to predict various internal flows such as flow in the diffuser and flow in a ribbed channel. The same Eulerian velocity correction method is extended to predict the 3-D laminar flows in various ducts. The steady state results have been compared with benchmark solutions and the agreement appears to be good.