Separating inviscid flows

Summary The separating flow of an inviscid fluid is not only a limit solution of the steady separating, laminar fluid flow at high values of the Reynolds number but it is also part of its structure (Smith [1], [2]). This work aims at reexamining the separating flow of inviscid fluid past a bluff body which is fixed in an otherwise uniform stream of fluid. For the purpose of this paper we will assume that the bluff body is a circular cylinder but the theory is applicable to bodies of any shape. It is further assumed that the fluid is in steady two-dimensional motion and is inviscid and of constant density. The flow structure is assumed to consist of a separated flow region, caviting flows in which there exists a free surface on which the pressure is constant, and a wake. A twin spiral vortex model is used in order to determine the shape of the free streamline. Based on the free streamline theory the problem reduces to solving a mixed boundary value problem and a Hilbert solution for the inverse problem in the auxiliary plane is obtained. When we consider the flow in the physical plane the problem is transformed into a direct problem in which the geometry of the solid body is given in advance. We assume that the separation is smooth and thus the curvature of the free streamline at the point of free detachment be equal to that of the body surface. A numerical method for solving the two-dimensional potential flows past arbitrarily shaped curved bluff bodies is developed.When the cavitation number is zero the angle of separation is approximately 55° and the computed results predict that the position of the separation point will move backward as the cavition number increases. The relationships between the drag coefficient, and the width and length of the cavity is determined and is found to be in very good agreement with the predictions of Smith [1].     

Circle theorems for inviscid steady flows

General circle theorems which localize the complex eigenfrequencies arising in the linear stability analysis of conservative steady flows are given. Howard's circle theorem for incompressible plane parallel flow is contained as a special case. Two applications are considered: swirling flow of an inviscid incompressible fluid, and rotating flow of an inviscid, incompressible, perfectly conducting magnetofluid with an axial magnetic field. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.     

The Oseen model for internal separated flows

Summary The flow field studied in this paper is the viscous laminar, separated flow downstream of a sudden expansion in a two-dimensional duct. The flow is modelled by the Oseen equations and a solution is sought for the downstream flow given the conditions at the sudden expansion. First, the exact solution to a high-Reynolds-number limit equation suggested by Kumar and Yajnik [6] is obtained. Next, the solution to the full equations is sought in terms of an eigenfunction-expansion procedure which leads to a non-standard eigenvalue problem. A detailed study is made of the latter and a number of expansion procedures are considered for the boundary-value problem. Specific calculations of the separated flow are presented for Reynolds numbers R=10 ⁿ , n=0–5. It is found that as R the solution of the full equation does indeed agree with the solution of the high-Reynolds-number limit equation. In particular it is found that the length of the recirculating region x _r scales with R as R.     

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.     

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.     

Small oscillations of inviscid incompressible stratified parallel and swirling flows

Existence theorems, minimax principles and comparison theorems are obtained for the real eigenfrequencies of inviscid, incompressible, stratified plane parallel flow and swirling flow.     

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.     

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.     

Viscous and inviscid matching of three-dimensional free-surface flows utilizing shell functions

A methodology is presented for matching a solution to a three-dimensional free-surface viscous flow in an interior region to an inviscid free-surface flow in an outer region. The outer solution is solved in a general manner in terms of integrals in time and space of a time-dependent free-surface Green function. A cylindrical matching geometry and orthogonal basis functions are exploited to reduce the number of integrals required to characterize the general solution and to eliminate computational difficulties in evaluating singular and highly oscillatory integrals associated with the free-surface Green-function kernel. The resulting outer flow is matched to a solution of the Navier?CStokes equations in the interior region and the matching interface is demonstrated to be transparent to both incoming and outgoing free-surface waves.     

Concerning closed-streamline flows with discontinuous boundary conditions

High-Reynolds-number (Re) flow containing closed streamlines (Prandtl-Batchelor flows), within a region enclosed by a smooth boundary at which the boundary conditions are discontinuous, is considered. In spite of the need for local analysis to account fully for flow at points of discontinuity, asymptotic analysis for Re 1 indicates that the resulting mathematical problem for determining the uniform vorticity ₀) in these situations, requiring the solution of periodic boundary-layer equations, is in essence the same as that for a flow with continuous boundary data. Extensions are proposed to earlier work [3] to enable ₀ to be computed numerically; these require coordinate transformations for the boundary-layer variables at singularities, as well as a two-zone numerical integration scheme. The ideas are demonstrated numerically for the classical circular sleeve.     

A numerical study on low and higher-order potential based BEM for 2D inviscid flows

A formal error analysis of the order of approximation of a potential based boundary element method (BEM) for two-dimensional flows is performed in order to derive consistent approximations for the potential integrals. Two higher-order approaches satisfying consistency requirements to attain second and third order convergence in the potential are selected for numerical implementation. From the formal local expansions of the potential integrals the influence coefficients are derived and evaluated analytically. In order to assess the methods accuracy, the low and higher-order methods are applied to two-dimensional steady flows around analytical foils. A numerical error analysis is done and a comparison between their theoretical and numerical asymptotic order of accuracy performed. The first author acknowledges the financial support granted by Fundação para a Ciência e a Tecnologia, Ph.D. grant PRAXISXXI/BD/2226/99. This work was done under the project PRAXIS/2/2.1/MAR/1723/95.     

Large-time behaviour of solutions of the inviscid non-planar Burgers equation

Large-time behaviour of the entropy solution of an initial-value problem (IVP) for the inviscid non-planar Burgers equation is studied. The initial profile is assumed to be non-negative, bounded and compactly supported. The large-time behaviour of the support function of this entropy solution is also presented. This is achieved via the construction of the entropy solution of the IVP for the inviscid non-planar Burgers equation subject to the top-hat initial condition using the method of characteristics, Rankine–Hugoniot jump condition, and a similarity solution.     

Variational properties of stationary inviscid incompressible flows with possible abrupt inhomogeneity or free surface

The inviscid flows, possibly rotational and nonsmooth, which satisfy the equation of stationary incompressible hydrodynamics, are characterized as giving zero variation rate to some real functional when the corresponding scalar and vector fields are transported by what is called a carrier, i.e. a mobile differential manifold. This transport does not have to preserve volumes; the Bernoulli function figures as the natural unknown scalar field rather than the pressure. Inhomogeneity may be sharp, implying in particular the presence of free surfaces. The key mathematical concept is that of a divergence-free vector measure convected by a carrier. For easier handling of this concept, some versions of the main variational statement are derived, involving vector potentials and stream functions in two or three dimensions; axially symmetric flows are also considered.     

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.     

Exact solutions for Couette and Poiseuille flows for fourth grade fluids

Summary Exact solutions for four types of flows between two parallel plates are presented, viz. Couette flow, plug flow, Poiseuille flow and generalized Couette flow. The nonlinear second-order ordinary differential equation for the velocity field is solved exactly in each case. These solutions are compared to those found by perturbation and homotopy analysis methods by Siddiqui et al. [1].