Flow Past Bodies Research Articles

Investigation of transonic bounded gas flow past three-dimensional slender bodies

The pressure coefficient distributions induced in the process of a bounded gas flow past a schematized vehicle of the “Khotol” type are obtained for various transonic regimes in a wind tunnel with a perforated working section as a result of simultaneously solving the boundary-value problems for the near and far flow fields. These distributions are compared with the pressure coefficient distributions in the case of unbounded flow past the body. The additional pressure coefficient induced by the tunnel walls is calculated for subsonic flow past a vehicle in a cylindrical wind tunnel of large radius. The calculation results are consistent with the asymptotic Malmuth theory for subsonic velocities.

STRUCTURAL CHANGES IN TRANSIENT STOKES FLOW

In his numerical study of oscillatory Stokes flow past particular axisymmetric bodies, Pozrikidis described the process of separation from the body together with the development and decay of eddies in the fluid. Here we extend his results in three different directions. First, the limiting situation is considered when the frequency of oscillations is small, with a description of the singular perturbation features of the resulting flow; the fundamental problem which describes the manner in which separation occurs is formulated in general, and solved for a spheroid. Secondly, some fairly general conditions are presented under which separation takes place from the surface of a sphere when the streaming motion has a variable velocity; these indicate that such a phenomenon should be seen as a common occurrence. Last, the flow due to an oscillatory stokeslet inside a sphere is considered, and it is found that the behaviour is effectively quasisteady for the complete cycle except for a brief period shortly after the reversal in the forcing term; even for moderate values of the frequency this transition occupies less than one per cent of the cycle

Simulation of hypersonic flows in chemical and thermodynamic nonequilibrium at low and moderate reynolds numbers

In studies of rarefied gas flow it is relevant to consider the limits of applicability of the continuum model. In problems of hypersonic flow past bodies the deviation from the continuum model, reflected in breakdown of the linear relationship between fluxes and "forces," is manifested primarily near the surface and within the head shock, the thickness of which is proportional to the Knudsen number. Since the equations of hydrodynamics are an asymptotic limit of solutions of Boltzmann equations at Knudsen numbers that tend to zero, in the absence of reliable experimental data we can naturally try to compare the solutions of the Navier-Stokes equations at small Reynolds numbers Re~ with the solution of Boltzmann equations. In [1] the solution has been obtained in the framework of the locally self-similar approximation of the Navier-Stokes equations and it has been established that the flow near the body is characterized by absence of equilibrium between translational and vibrational degrees of freedom. All vibrational temperatures of the molecular components were assumed equal in this study. In the framework of parabolized Navier-Stokes equations, which are an asymptotic form of the Navier-Stokes equations retaining terms of order O(1) and O(Re~) [6], a numerical solution has been obtained for the two-dimensional axisymmetric problem of flow past a hyperboloid of revolution, which approximately models the flow in the neighborhood of the stagnation line of a reentrant spacecraft in the range of altitudes from 85 to 110 kin. The method of global iterations by the pressure gradient is used; this method was previously proposed in [2] for homogeneous gas flows. In this article, we consider a model with different temperatures of the molecular components, which contrary to [3] allows for feedback from the chemical reactions to vibrational relaxation. The boundary slip conditions include a jump of vibrational energy at the surface. The calculations are compared with direct Monte-Carlo simulation results [4] obtained under the same conditions. For flows of a chemically reactive gas, this comparison has been previously carried out ignoring vibrational relaxation [4, 5]. We consider stationary supersonic flow past a smooth blunt body immersed in a viscous dissociated air mixture in chemical and thermodynamic nonequilibrium. In the coordinate system attached to the surface of the immersed body (the coordinate x is measured along the surface and y along the normal to the surface) we write the following equations [7]: the equation of continuity; two projections of the momentum equation; the equation of heat influx, with explicit terms associated with relaxation of vibrational degrees of freedom (as thermodynamic equilibrium is approached, the additional terms vanish and the heat influx equation takes the standard form, with vibrational and translational degrees of freedom in equilibrium); the equation of balance of vibrational energy for each species of molecules; the equation obtained by summation of the equations of balance of vibrational energy for each species of molecules assuming equal vibrational temperatures; equations of diffusion of the reaction products; equations of diffusion of chemical elements closing these equations; Stefan-Maxwell equations; state equation. When specifying boundary conditions on the surface of the impermeable body, we allow for slip effects, temperature discontinuity, and catalytic recombination of atoms on the wall [8]. For the equations of vibrational energy balance of each species of molecules, new boundary conditions are proposed that allow for the discontinuity of the vibrational energy on the surface of the body. Conditions corresponding to the impinging flow parameters are specified on the outer boundary. Five components are assumed present in the perturbed flow region: N 2, 02, NO, N, O. All these components participate in dissociation, recombination, and exchange reactions.

Three-dimensional vortical flow past bodies of revolution with conical afterbodies

Three-dimensional separated flows past bodies of revolution with conical afterbodies are studied both experimentally and numerically. The sudden increase in drag force due to the small increase in the semi-apex angle of the conical afterbody which is related to the critical geometry is revealed by force measurement and flow visualization. The details of the associated physical phenomenon, i. e. 3D separation and vortex shedding is elucidated by numerical simulations based on the finite-volume solution method for the Navier-Stokes equation in the curvilinear, boundary-fitted grid system. It is shown by the computer graphic drawings that the large-scale structure of separated vortices is composed of ring vortices and longitudinal vortices originating from the center of the ring vortices.

On the applicability of some approximate similitude laws in hypersonic aerodynamics

The range of applicability of some similitude laws for heat transfer, friction and drag coefficients is discussed on the basis of numerical solutions of the complete viscous shock layer equations describing hypersonic flow past blunt bodies.

Interactive, Animated Visualization Environment for Three-Dimensional Fluid Flow

We describe an interactive, animated graphics environ ment that can be used to visualize the results of com putational fluid dynamics simulations. With this sys tem, the user can visualize flow past bodies or in chambers, using a variety of techniques borrowed from laboratory experiments, including dye injection, smoke visualization, and bubble wires. The environ ment provides the capability of three-dimensional rota tion of the flow under study, as well as placement of internal flow probes that can read location, velocity, and secondary quantities (such as vorticity and pres sure), which can be calculated from available data as the visualization unfolds. The system is implemented on a Connection Machine 2 attached to a framebuffer. As input, it accepts discretized data from a three- dimensional fluid flow calculation. Data should be provided on a three-dimensional grid at discrete time steps, and internal bodies and flow boundaries are de scribed using discrete panels. Input parameters are menu-driven, and images are updated at five frames per second. A prototype, second version of the system is implemented on a Connection Machine 5 using the Application Visualization System.

On the simulation of turbulent flow past bluff bodies

On the simulation of turbulent flow past bluff bodies

Numerical modeling of three-dimensional viscous flows using a space-marching method with pressure iterations

A method of calculating steady three-dimensional compressible and incompressible flows on the basis of simplified Navier-Stokes equations is examined. Some calculation results are presented. These relate to the problem of supersonic flow past blunt bodies and to incompressible channel flows.

Supersonic viscous flow past two-dimensional blunt-nosed bodies

Supersonic viscous flow past two-dimensional blunt-nosed bodies

Comparison of gas-dynamic models for hypersonic flow past bodies

Various gas-dynamic models for describing chemically non-equilibrium flows are compared using the example of the steady flow past the blunt nose of the “Buran” [1] and “Space Shuttle” vehicles during their descent from orbit. Models of locally self-similar approximations of the Navier-Stokes equations [2], of a chemically equilibrium and non-equilibrium complete viscous shock layer (CVSL) [3] and a model of a thin viscous shock layer (TVSL) [4] are considered. In all the models the occurrence of physicochemical processes was taken into account in the same way using fixed values of the constants for the gas-phase chemical reactions (their effect has been considered in [5]). Good agreement between the results of calculations of the heat flux at the critical point is found. Chemically non-equilibrium flows have been considered earlier using the approximate Navier-Stokes equations [6], within the framework of a TVSL [7] and a CVSL [8, 9](for more detail, see the review [10]). The TVSL and CVSL models were compared in [11] in the case of flows of a uniform gas.

Axisymmetric analogy for three-dimensional viscous flow problems

A method of solving three-dimensional flow problems with the aid of two-dimensional solutions, which can be used for any Reynolds numbers, is proposed. The method is based on the use of similarity relations obtained in the theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer. These relations express the heat flux and the friction stress on the lateral surface of a three-dimensional body in terms of the values on the surface of an axisymmetric body. The accuracy is estimated by comparing the results with those of a numerical finite-difference calculation of the flow past bodies of various shapes. Similar similarity relations were previously obtained in [1] for the plane of symmetry of a blunt body.

Problems of incompressible flow past thin blades and correction of their shape

Solutions to the problem of the flow of an ideal fluid past a blade or cascade of blades with low blocking factor are found in the framework of the first approximation of the theory of perturbations of the flow past infinitely thin arcs. Problems of correction of the shape of the blades are also considered. Problems associated with the application of perturbation theory in problems of flow past bodies are discussed in Van Dyke's monograph [1]. The present paper includes an example of realization of this theory for the thin curved blades that are widely used in compressor construction. Searches for effective methods for calculating the shape of blades to ensure necessary gas-dynamic properties, for example, a given distribution on the blades of the velocity of separationless flow, led to the appearance of algorithms based on the theory of small perturbations for a thin wing of finite span [2] and a single airfoil in a gas flow [3]. In such an approach, the problem of constructing the required profile can be formulated as a sequence of corrections of the boundary of the flow region with respect to small variations of the boundary values of the flow velocity. The paper contains a general formulation of the linear problem of the correction of the flow boundary, an algorithm for its solution in the case of thin blades in an incompressible flow, and analysis of the obtained solutions. Examples of calculations are presented.

The local linearization boundary element method with application to high subsonic flow past body

The boundary element method is investigated for the computation of high subsonic flow about three-dimensional bodies with arbitrary shape. Based on the nonlinear transonic small-disturbance equation, an integral equation is obtained by the local linearization technique and Green's function method, and solved by boundary element method. The local linearization parameters are determined iteratively. Computations performed for both axisymmetric and nonaxisymmetric configurations show that the iteration converges rapidly and the results are satisfying. The combination of boundary element method and local linearization technique seems to be an effective engineering approach to solve some nonlinear equations with complex boundaries.

Computation of unsteady flow past blunt bodies

In accordance with the recent experimental research for flow visualization, the unsteady behavior of the starting period is investigated numerically for flow past blunt bodies. Finite difference methods are employed to solve the unsteady two-dimensional incompressible Navier-Stokes equations. A short discussion is presented of explicit, implicit and ADI methods. Finally, the explicit and ADI schemes are used to study the flow field in the starting period for flow past mountain-shaped and rectangular bodies.

The separated flow past a cylinder in a rotating frame

In this paper a numerical model for the viscous flow past a cylinder in a rotating frame is discussed when both the Rossby number Ro and Ekman number E are small. The results of this model are analysed and compared to an inviscid study by Page (1987) applicable in the limit E → 0 with Ro = O(E½). The detailed structure of the separated flow is also examined and compared to the proposals for the higher-order flow in E¼ layers in Page (1987) which were based, in part, on the theory of Smith (1979, 1985) for the non-rotating flow past bluff bodies. Some discrepancies between this theory and the numerical results are noted.

Application of the boundary integral equation methods to subsonic flow past bodies and wings

In this paper the boundary integral equation method is applied to the subsonic flow in the presence of three-dimensional bodies or of wings. The integral equation is obtained by the aid of a source distribution on the body surface. The equation discretization is obtained via a collocation method using triangular elements. Numerical tests are performed for the case of the sphere in incompressible fluid (in this case the exact solution is known). Numerical results are also given for the ellipsoid at 0° and 10° incidence for various values of the Mach numberM. The method allows also to consider lifting wings. Numerical determinations are performed for a wing whose section is a NACA64A-008 airfoil.

Investigation of supersonic radiative flow past bodies at low altitudes

The radiative heat transfer problem for bodies traveling through the earth's atmosphere, as formulated in [1, 3], is considered in the case of low altitudes, i.e., for high gas densities and optical thicknesses in the shock layer. These factors require the use of improved methods of calculation.

A numerical analysis of the unsteady flow past bluff bodies

The paper describes in detail a relatively sophisticated numerical approach, using the Boundary Element Method in conjunction with the Discrete Vortex Model, to represent the complex unsteady flow field around a bluff body with separating shear layers. Important steps in the numerical analysis of this challenging problem are discussed and a performance evaluation algorithm established. Of considerable importance is the effect of computational parameters such as number of elements representing the geometry, time-step size, location of the nascent vortices, etc., on the accuracy of results and the associated cost.

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].

Similarity of three-dimensional and axisymmetric chemically-nonequilibrium flows in the neighborhood of a plane of symmetry

Three-dimensional chemically-nonequilibrium flow past blunt bodies in the neighborhood of the plane of symmetry is investigated within the framework of viscous shock layer theory. The similarity of three-dimensional and axisymmetric flows, previously established in [1] for a uniform gas, is extended to chemically-nonequilibrium gas flows. It is shown that the problem of determining the heat fluxes and friction stress in the neighborhood of the line of flow divergence can be reduced to the problem of determining these quantities for the axisymmetric body. The validity of the axisymmetric analogy is verified by carrying out numerical calculations for bodies of different shapes re-entering the earth's atmosphere along a gliding trajectory. Various models of surface catalytic activity are considered. The use of similarity relations makes it possible to apply existing programs for calculating axisymmetric flows to the solution of three-dimensional problems.