The unsteady flow of a weakly compressible fluid in a thin porous layer II: three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that $\epsilon\propto h/l\ll 1$, we show that, to leading order in $\epsilon$, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in $\epsilon$ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, $\epsilon$, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

The unsteady flow of a weakly compressible fluid in a thin porous layer III: three-dimensional computations

We describe a novel method for determining the pressure and velocity fields for a weakly compressible fluid flowing in a thin three-dimensional layer composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Our approach uses the method of matched asymptotic expansions to derive expressions for all significant process quantities, the computation of which requires only the solution of linear, elliptic, two-dimensional boundary value and eigenvalue problems. In this article, we provide full implementation details and present numerical results demonstrating the efficiency and accuracy of our scheme.

Linear instability of orthogonal compressible leading-edge boundary layer flow

Instability analysis of compressible orthogonal swept leading-edge boundary layer flow was performed in the context of BiGlobal linear theory. 1, 2 An algorithm was developed exploiting the sparsity characteristics of the matrix discretizing the PDE-based eigenvalue problem. This allowed use of the MUMPS sparse linear algebra package 3 to obtain a direct solution of the linear systems associated with the Arnoldi iteration. The developed algorithm was then applied to efficiently analyze the effect of compressibility on the stability of the swept leading-edge boundary layer and obtain neutral curves of this flow as a function of the Mach number in the range 0 ≤ Ma ≤ 1. The present numerical results fully confirmed the asymptotic theory results of Theofilis et al. 4 Up to the maximum Mach number value studied, it was found that an increase of this parameter reduces the critical Reynolds number and the range of the unstable spanwise wavenumbers.

Global Stability Analysis of a Compressible Turbulent Flow around a High-Lift Configuration

The purpose of this work is to analyze a complex high lift configuration for which significant regions of separated flow are present. Current state of the art methods have some diffculty to predict the origin and the progression of this separated flow when increasing the angle of attack. The mechanisms responsible for the maximum lift limit on multi-element wing con?gurations are not clear; this stability analysis could help to understand the physics behind the phenomenon and to find a relation between the flow separation and the instability onset. The methodology presented herein consists in the computation of a steady base flow solution based on a finite volume discretization and a proposal of the solution for a generalized eigenvalue problem corresponding to the perturbed and linearized problem. The eigenvalue problem has been solved with the Arnoldi iterative method, one of the Krylov subspace projection methods. The described methodology was applied to the NACA0012 test case in subsonic and in transonic conditions and, finally, for the first time to the authors knowledge, on an industrial multi-component geometry, such as the A310 airfoil, in order to identify low frequency instabilities related to the separation. One important conclusion is that for all the analyzed geometries, one unstable mode related to flow separation appears for an angle of attack greater than the one correspondent to the maximum lift coe?cient condition. Finally, an adjoint study was carried out in order to evaluate the receptivity and the structural sensitivity of the geometries, giving an indication of the domain region that could be modified resulting in the biggest change of the flowfield.

Bibliography on flow of compressible fluids,

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Unsteady compressible second-order boundary layers at the stagnation point of two-dimensional and axisymmetric bodies

The unsteady laminar compressible boundary-layer flow over two-dimensional and axisymmetric bodies at the stagnation point with mass transfer has been studied for all second-order boundary layer effects when the basic potential flow admits selfsimilarity. The solutions for the governing equations are obtained by using an implicit finite-difference scheme. Computations have been carried out for different values of the parameters characterizing the unsteadiness in the free stream velocity, wall temperature, mass transfer rate and variable gas properties. The results are found to be strongly affected by the unsteadiness in the free stream velocity. For large injection rates the second-orderboundary layer effects may prevail over the first-order boundary layer, but reverse is true for suction. The wall temperature and the variation of the density-viscosity product across the boundary layer appreciably change the skin-friction and heat-transfer rates due to second-order boundary-layer effects.

Self-similar solution of unsteady compressible three-dimensional stagnation-point boundary layers

The self-similar solution of the unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point with mass transfer has been obtained when the free-stream velocity varies inversely as a linear function of time. The resulting ordinary differential equations have been solved numerically using an implicit finite-difference scheme. The results are found to be strongly dependent on the parameter characterizing the unsteadiness in the free-stream velocity. The velocity profiles show some features not encountered in steady flows.

Unsteady nonsimilar compressible laminar two-dimensional and axisymmetric boundary-layer flows

Unsteady nonsimilar laminar compressibletwo-dimensional and axisymmetric boundarylayer flows have been studied when external velocity varies arbitrarily with time and the flow is nonhomentropic. The governing nonlinear partial differential equations with three independent variables have been solved using an implicit finite difference scheme with quasilinearization technique from the origin to the point of zero skin-friction. The results have been obtained for (i) an accelerating stream and (ii) a fluctuating stream. The skin friction responds to the fluctuations in the free stream more compared to the heat transfer. It is observed that Mach number and hot wall cause the point of zero skin friction to occur earlier whereas cold wall delays it.

Long waves in inviscid compressible atmosphere II

Solitary waves have been found in an adiabatic compressible atmosphere which, in ambient state, has winds and temperature gradient, generalizing our earlier results for the isothermal atmosphere. Explicit results are obtained for the special case of linear temperature and linear wind distributions in the undisturbed conditions. An important result of the study is that the number of possible critical speeds of the flow depends crucially on whether the maximum Richardson number (which is variable in the present example) is greater or less than 1/4.

Laminar compressible boundary layers with vectored mass transfer

The paper studies the influence of vectored suction or injection on the flow and heat transfer at the stagnation point of a two-dimensional body (a cylinder) and an axisymmetric body (a sphere) with allowance for the effects of variable gas properties. The analysis is based on the boundary-layer equations in dimensionless form for the steady compressible fluid with variable properties in the stagnation region of a two-dimensional or an axisymmetric body with tangential and normal surface mass transfer under similarity requirements. It is shown that the variation of the density-viscosity product across the boundary layer has a strong effect on the skin friction and heat transfer. This gives rise to a point of inflection which can be removed by suction and by increasing the wall temperature. The skin friction and heat transfer are significantly affected by the pressure gradient parameter.

Compressible magnetohydrodynamic boundary layer swirling nozzle and diffuser flows with a magnetic field

The flow, heat and mass transfer problem for boundary layer swirling flow of a laminar steady compressible electrically conducting gas with variable properties through a conical nozzle and a diffuser with an applied magnetic field has been studied. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme after they have been transformed into dimensionless form using the modified Lees transformation. The results indicate that the skin friction and heat transfer strongly depend on the magnetic field, mass transfer and variation of the density-viscosity product across the boundary layer. However, the effect of the variation of the density-viscosity product is more pronounced in the case of a nozzle than in the case of a diffuser. It has been found that large swirl is required to produce strong effect on the skin friction and heat transfer. Separationless flow along the entire length of the diffuser can be obtained by applying appropriate amount of suction. The results are found to be in good agreement with those of the local nonsimilarity method, but they differ quite significantly from those of the local similarity method.

Boundary-layer effects on internal flow choking in dual-thrust solid rocket motors

Theoretical studies have been carried out to examine internal flow choking in the inert simulators of a dual-thrust motor. Using a two-dimensional k-omega turbulence model, detailed parametric studies have been carried out to examine aerodynamic choking and the existence of a fluid throat at the transition region during the startup transient of dual-thrust motors. This code solves standard k-omega turbulence equations with shear flow corrections using a coupled second-order-implicit unsteady formulation. In the numerical study, a fully implicit finite volume scheme of the compressible, Reynolds-averaged, Navier-Stokes equations is employed. It was observed that, at the subsonic inflow conditions, there is a possibility of the occurrence of internal flow choking in dual-thrust motors due to the formation of a fluid throat at the beginning of the transition region induced by area blockage caused by boundary-layer-displacement thickness. It has been observed that a 55% increase in the upstream port area of the dual-thrust motor contributes to a 25% reduction in blockage factor at the transition region, which could negate the internal How choking and supplement with an early choking of the dual-thrust motor nozzle. If the height of the upstream port relative to the motor length is too small, the developing boundary layers from either side of the port can interact, leading to a choked,flow. On the other hand, if the developing boundary layers are far enough apart, then choking does not occur. The blockage factor is greater in magnitude for the choked case than for the unchoked case. More tangible explanations are presented in this paper for the boundary-layer blockage and the internal flow choking in dual-thrust motors, which hitherto has been unexplored.

On compressible boundary layer on a flat plate with uniform suction

A quartic profile in terms of the normal distance from the wall has been taken and coefficients are evaluated by satisfying one more boundary condition on the wall than the usual one. By doing so, the limitations about the Reynolds number of the quartic profile adopted by Lew (1949) has been removed. The Kármán (1921) Momentum Integral Equation has been used to evaluate the various characteristics of the flow. A comparative study of Lew's quartic profile and exponential profile together with the quartic profile of the present paper has been undertaken and the graphs for the various characteristics of the flow for a number of Mach numbers and suction coefficients have been drawn. At the end, certain conclusions of general nature about the velocity profiles have been recorded.