Computation of compressible quasi-axisymmetric slender vortex flow and breakdown

The unsteady, compressible Navier-Stokes equations are used to compute and analyze compressible quasi-axisymmetric isolated vortices. The Navier-Stokes equations are solved using an implicit, upwind, flux difference splitting finite volume scheme. The developed three dimensional solver was verified by comparing its solution profiles with those of a slender, quasi-axisymmetric vortex solver for a subsonic, quasi-axisymmetric vortex in an unbounded domain. The Navier-Stokes solver is then used to solve for a supersonic, quasi-axisymmetric vortex flow in a configured circular duct. Steady and unsteady vortex-shock interactions and breakdown were captured. The problem was also calculated using the Euler solver of the same code; the results were compared with those of the Navier-Stokes solver. The effect of the initial swirl was investigated

Supersonic quasi-axisymmetric vortex breakdown

An extensive computational study of supersonic quasi-axisymmetric vortex breakdown in a configured circular duct is presented. The unsteady, compressible, full Navier-Stokes (NS) equations are used. The NS equations are solved for the quasi-axisymmetric flows using an implicit, upwind, flux difference splitting, finite volume scheme. The quasi-axisymmetric solutions are time accurate and are obtained by forcing the components of the flowfield vector to be equal on two axial planes, which are in close proximity of each other. The effect of Reynolds number, for laminar flows, on the evolution and persistence of vortex breakdown, is studied. Finally, the effect of swirl ration at the duct inlet is investigated

Analysis and mitigation of numerical dissipation in inviscid and viscid computation of vortex-dominated flows

The conservative unsteady Euler equations for the flow relative motion in the moving frame of reference are used to solve for the steady and unsteady flows around sharp-edged delta wings. The resulting equations are solved by using an implicit approximately-factored finite volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in cross flow planes at the grid chord stations. The computational applications cover a steady flow around a delta wing whose results serve as the initial conditions for the unsteady flow around a pitching delta wing about a large angle of attack. The steady results are compared with the experimental data and the periodic solution is achieved within the third cycle of oscillation

Applications of perturbation techniques

Two perturbation techniques were applied to two singular perturbation problems in heat transfer to obtain uniformly valid solutions which can serve as benchmarks for finite difference and finite element techniques. In the first problem, the method of strained parameters coupled with the application of a solvability condition is used to obtain a uniform solution for the problem of unsteady heat conduction in a long nearly circular cylinder. In the second problem, the method of matched asymptotic expansion coupled with Van Dyke's matching principle is used to obtain a uniform solution for the problem of one dimensional conduction-convection heat transfer of a uniform fluid flow

Unsteady hybrid vortex technique for transonic vortex flows and flutter application

Papers resulting from work performed from January 1, 1987 to July 31, 1987 are listed. Transonic computational schemes based on Integral Equation Formulation of the full potential equation were presented. Classical and zero-total pressure-loss sets of Euler equations applied to delta wings were examined

Computational technique for compressible vortex flows using the integral equation solution

The steady full-potential equation is written in the form of Poisson's equation, and the solution for the velocity field is expressed in terms of an integral equation. The integral solution consists of two surface integrals and one volume integral. The solution is obtained through successive iteration cycles. Each cycle of iteration consists of two sub-cycles, an inner cycle for wake relaxation and an out cycle for the strength of the source distribution integrals representing the flow compressibility. The density gradients in the source distribution is computed by using a type-differencing scheme of the Murman-Cole type. The method is applied to delta wings and the numerical examples show that a curved shock is captured on the wing suction side beneath the leading edge vortex sheet. Recently, a modified version of the scheme was applied to rectangular wings. In this modified scheme, the surface integral terms were computed by using a bilinear distribution of vorticity on triangular vortex panels which represent the wing and its wake. The results were compared with the available experimental data and they are in good agreement

Transonic airfoil computation using the integral equation with and without embedded Euler domains

Two transonic computational schemes which are based on the Integral Equation Formulation of the full potential equation were presented. The first scheme is a Shock Capturing-Shock Fitting (SCSF) scheme which uses the full potential equation throughout with the exception of the shock wave where the Rankine-Hugoniot relations are used to cross and fit the shock. The second scheme is an Integral Equation with Embedded Euler (IEEE) scheme which uses the full potential equation with an embedded region where the Euler equations are used. The two schemes are applied to several transonic airfoil flows and the results were compared with numerous computational results and experimental domains with fine grids. The SCSF-scheme is restricted to flows with weak shock, while the IEEE-scheme can handle strong shocks. Currently, the IEEE scheme is applied to other transonic flows with strong shocks as well as to unsteady pitching oscillations

Application of the nonlinear vortex-lattice concept to aircraft-interference problems

A discrete-vortex model was developed to account for the hazardous effects of the vortex trail issued from the edges of separation of a large leading wing on a small trailing wing. The model is divided into three main parts: the leading wing and its near wake, the near and far wakes of the leading wing, and the trailing wing and the portion of the far wake in its vicinity. The normal force, pitching moment, and rolling moment coefficients for the trailing wing are calculated. The circulation distribution in the vortex trail is calculated in the first part of the model where the leading wing is far upstream and hence is considered isolated. A numerical example is solved to demonstrate the feasibility of using this method to study interference between aircraft. The numerical results show the correct trends: The following wing experiences a loss in lift between the wing-tip vortex systems of the leading wing, a gain outside this region, and strong rolling moments which can change sign as the lateral relative position changes. All the results are strongly dependent on the vertical relative position

Three dimensional steady and unsteady asymmetric flow past wings of arbitrary planforms

The nonlinear discrete vortex method was extended to treat the problem of asymmetric flows past a wing with leading-edge separation, including steady and unsteady flows. The problem was formulated in terms of a body-fixed frame of reference, and the nonlinear discrete vortex method was modified accordingly. Only examples of flows past delta wings are presented. Comparison of these results with experimental results for a delta wing undergoing a steady rolling motion at zero angle of attack demonstrates the superiority of the present method in obtaining highly accurate loads. Numerical results for yawed wings at large angles of attack are also presented. In all cases, total load coefficients, pressure distributions and shapes of the free-vortex sheets are shown

Navier-Stokes Simulation of Quasi-Axisymmetric and Three-Dimensional Supersonic Vortex Breakdown

Computational simulation of supersonic vortex breakdown is considered for internal and external flow applications. The interaction of a supersonic swirling flow with a shock wave in bounded and unbounded domains is studied. The problem is formulated using the unsteady, compressible, full Navier-Stokes equations which are solved using an implicit, flux-difference splitting, finite-volume scheme. Solutions are obtained for quasi-axisymmetric and three-dimensional flows. The quasi-axisymmetric solutions are obtained by forcing the components of the flowfield vector to be equal on two axial planes, which are in close proximity to each other. For the flow in a bounded domain, a supersonic swirling flow is introduced into a configured circular duct. The duct is designed such that a shock wave intersects with the incoming swirling flow in the inlet portion. For the quasi-axisymmetric flow problem, a parametric study is performed which includes the effects of the Reynolds number, Mach number, swirl ratio and the type of exit-boundary conditions on the development and behavior of vortex breakdown. The effect of the duct wall boundary-layer flow on the vortex breakdown is also investigated. For the same duct geometry, three-dimensional effects are studied along with the effect of the duct wall boundary-layer flow. For the external flow application, a supersonic swirling jet is issued from a nozzle into a uniform supersonic flow of lower Mach number. For the quasi-axisymmetric flow problem, the effects of the Reynolds number and the type of downstream-boundary conditions are studied. For the three-dimensional flow problem, the effects of the grid fineness, grid-point distribution, grid shape and swirl ratio on the vortex breakdown are studied. The results show several modes of vortex breakdown such as no-breakdown, transient single-bubble breakdown, transient multi-bubble breakdown, periodic multi-bubble multi-frequency breakdown and helical spiral breakdown. In another application, a subsonic steady quasi-axisymmetric flow of an isolated slender vortex core is considered. The solution is obtained using a simple set of parabolic equations. The results are in excellent agreement with those of the full Navier-Stokes equations