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All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

New time scale based k-epsilon model for near-wall turbulence

A k-epsilon model is proposed for wall bonded turbulent flows. In this model, the eddy viscosity is characterized by a turbulent velocity scale and a turbulent time scale. The time scale is bounded from below by the Kolmogorov time scale. The dissipation equation is reformulated using this time scale and no singularity exists at the wall. The damping function used in the eddy viscosity is chosen to be a function of R(sub y) = (k(sup 1/2)y)/v instead of y(+). Hence, the model could be used for flows with separation. The model constants used are the same as in the high Reynolds number standard k-epsilon model. Thus, the proposed model will be also suitable for flows far from the wall. Turbulent channel flows at different Reynolds numbers and turbulent boundary layer flows with and without pressure gradient are calculated. Results show that the model predictions are in good agreement with direct numerical simulation and experimental data.

Yang, Z.↗
NASA NTRS1993

A Galilean and tensorial invariant k-epsilon model for near wall turbulence

A k-epsilon model is proposed for wall bounded turbulent flows. In this model, the eddy viscosity is characterized by a turbulent velocity scale and a turbulent time scale. The time scale is bounded from below by the Kolmogorov time scale. The dissipation rate equation is reformulated using this time scale and no singularity exists at the wall. A new parameter R = k/S(nu) is introduced to characterize the damping function in the eddy viscosity. This parameter is determined by local properties of both the mean and the turbulent flow fields and is free from any geometry parameter. The proposed model is then Galilean and tensorial invariant. The model constants used are the same as in the high Reynolds number Standard k-epsilon Model. Thus, the proposed model will also be suitable for flows far from the wall. Turbulent channel flows and turbulent boundary layer flows with and without pressure gradients are calculated. Comparisons with the data from direct numerical simulations and experiments show that the model predictions are excellent for turbulent channel flows and turbulent boundary layers with favorable pressure gradients, good for turbulent boundary layers with zero pressure gradients, and fair for turbulent boundary layer with adverse pressure gradients.

Yang, Z.↗
NASA NTRS2005

Computation of Sound Propagation by Boundary Element Method

This report documents the development of a Boundary Element Method (BEM) code for the computation of sound propagation in uniform mean flows. The basic formulation and implementation follow the standard BEM methodology; the convective wave equation and the boundary conditions on the surfaces of the bodies in the flow are formulated into an integral equation and the method of collocation is used to discretize this equation into a matrix equation to be solved numerically. New features discussed here include the formulation of the additional terms due to the effects of the mean flow and the treatment of the numerical singularities in the implementation by the method of collocation. The effects of mean flows introduce terms in the integral equation that contain the gradients of the unknown, which is undesirable if the gradients are treated as additional unknowns, greatly increasing the sizes of the matrix equation, or if numerical differentiation is used to approximate the gradients, introducing numerical error in the computation. It is shown that these terms can be reformulated in terms of the unknown itself, making the integral equation very similar to the case without mean flows and simple for numerical implementation. To avoid asymptotic analysis in the treatment of numerical singularities in the method of collocation, as is conventionally done, we perform the surface integrations in the integral equation by using sub-triangles so that the field point never coincide with the evaluation points on the surfaces. This simplifies the formulation and greatly facilitates the implementation. To validate the method and the code, three canonic problems are studied. They are respectively the sound scattering by a sphere, the sound reflection by a plate in uniform mean flows and the sound propagation over a hump of irregular shape in uniform flows. The first two have analytical solutions and the third is solved by the method of Computational Aeroacoustics (CAA), all of which are used to compare the BEM solutions. The comparisons show very good agreements and validate the accuracy of the BEM approach implemented here.

Guo, Yueping↗
NASA NTRS1990

On a class of unsteady three-dimensional Navier Stokes solutions relevant to rotating disc flows: Threshold amplitudes and finite time singularities

A class of exact steady and unsteady solutions of the Navier Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x,-y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large, Karman's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic equilibrium flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Karman's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong, the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

Hall, Philip↗
NASA NTRS1983

Elasticity solutions for a class of composite laminate problems with stress singularities

A study on the fundamental mechanics of fiber-reinforced composite laminates with stress singularities is presented. Based on the theory of anisotropic elasticity and Lekhnitskii's complex-variable stress potentials, a system of coupled governing partial differential equations are established. An eigenfunction expansion method is introduced to determine the orders of stress singularities in composite laminates with various geometric configurations and material systems. Complete elasticity solutions are obtained for this class of singular composite laminate mechanics problems. Homogeneous solutions in eigenfunction series and particular solutions in polynomials are presented for several cases of interest. Three examples are given to illustrate the method of approach and the basic nature of the singular laminate elasticity solutions. The first problem is the well-known laminate free-edge stress problem, which has a rather weak stress singularity. The second problem is the important composite delamination problem, which has a strong crack-tip stress singularity. The third problem is the commonly encountered bonded composite joints, which has a complex solution structure with moderate orders of stress singularities.

Wang, S. S.↗
NASA NTRS1987

Experience in distributed parameter modeling of the Spacecraft Control Laboratory Experiment (SCOLE) structure

The Spacecraft Control Laboratory Experiment (SCOLE) configuration is used to compare exact and approximate solutions of the partial differential equations which define its structural dynamics. The need for a proof model for evaluating competing control laws demands that solutions be generated which not only exhibit accurate modal characteristics, but precise static deflections as well. Because precise pointing is required, the motion of the end bodies of the Shuttle-attached antenna must be known with great accuracy. Modal models are attractive because of their stable solutions but require hundreds of modes to obtain a static deflection accuracy of only one percent. Although proportional damping in bending agrees well with experimental results using the SCOLE experimental apparatus, modes which involve both torsion and bending differ significantly from proportional damping. A lumped mass model is used to generate exact static deflections, but only approximate modal characteristics. Asymptotic solutions to the distributed parameter system approximate very accurately the modal characteristics at high mode numbers. Ways are examined for refining the approximate solutions by applying a first-order variation and by employing singular perturbation techniques which are usually limited to ordinary differential equations. The most accurate solutions of the distributed parameter model of SCOLE are obtained by combining exact and asymptotic solutions.

Taylor, L. W.↗
NASA NTRS1984

Implementation of uniform perturbation method for potential flow past axisymmetric and two-dimensional bodies

The aerodynamic characteristics of potential flow past an axisymmetric slender body and a thin airfoil are calculated using a uniform perturbation analysis method. The method is based on the superposition of potentials of point singularities distributed inside the body. The strength distribution satisfies a linear integral equation by enforcing the flow tangency condition on the surface of the body. The complete uniform asymptotic expansion of its solution is obtained with respect to the slenderness ratio by modifying and adapting an existing technique. Results calculated by the perturbation analysis method are compared with the existing surface singularity panel method and some available analytical solutions for a number of cases under identical conditions. From these comparisons, it is found that the perturbation analysis method can provide quite accurate results for bodies with small slenderness ratio. The present method is much simpler and requires less memory and computation time than existing surface singularity panel methods of comparable accuracy.

Wong, T. C.↗
NASA NTRS1972

Equations of motion in the linear approximation.

Attempt to develop a gauge-invariant theory of the motion of singularities in an n-dimensional Riemannian space. A gauge-invariant theory is developed for a nongeodetic particle represented by a time-like world line in the linearized field theory. The solution of the basic algebraic and differential equations is to be constructed from the source line, the future null cones emanating from it, a scale factor, and nothing else, and the leading term of the solution is the Schwarzschild field. A class of possible solutions is examined, and it is shown that this class contains just one acceptable member - namely, a particle in constant acceleration.

Robinson, I.↗
NASA NTRS1973

Laminar film boiling on inclined isothermal flat plates.

Laminar film boiling from an inclined flat plate has been investigated analytically. Using the singular perturbation scheme, the complete set of Navier-Stokes equations is solved. The zeroth-order perturbation coinciding with the boundary-layer equations for vertical flat plates governs the problem. The higher-order perturbations become important near the leading edge and for large values of the inclination angle from the vertical. The assumption of zero interfacial velocity shows that, except for fluids having large (rho x mu) ratios, the results can be predicted using the vertical flat plate results by defining a modified Grashof parameter containing a cos phi term. When the interfacial shear is considered, the solutions indicate that for fluids having large (rho x mu) ratios, the heat transfer rates will be larger (approximately 15% maximum) than those predicted by the simplified model using zero interfacial velocity. In general, the inclination decreases the rate of heat transfer as well as the rate of evaporation.

Nagendra, H. R.↗
NASA NTRS1976

Solution of linear systems by a singular perturbation technique

An approximate solution is obtained for a singularly perturbed system of initial valued, time invariant, linear differential equations with multiple boundary layers. Conditions are stated under which the approximate solution converges uniformly to the exact solution as the perturbation parameter tends to zero. The solution is obtained by the method of matched asymptotic expansions. Use of the results for obtaining approximate solutions of general linear systems is discussed. An example is considered to illustrate the method and it is shown that the formulas derived give a readily computed uniform approximation.

Ardema, M. D.↗
NASA NTRS1977

Interacting turbulent boundary layer over a wavy wall

The two dimensional supersonic flow of a thick turbulent boundary layer over a train of relatively small wave-like protuberances is considered. The flow conditions and the geometry are such that there exists a strong interaction between the viscous and inviscid flow. The problem cannot be solved without inclusion of interaction effects due to the occurrence of the separation singularity in classical boundary layer methods. The interacting boundary layer equations are solved numerically using a time-like relaxation method with turbulence effects represented by the inclusion of the eddy viscosity model. Results are presented for flow over a train of up to six waves for Mach numbers of 10 and 32 million/meter, and wall temperature rations (T sub w/T sub 0) of 0.4 and 0.8. Limited comparisons with independent experimental and analytical results are also given. Detailed results on the influence of small protuberances on surface heating by boundary layers are presented.

Polak, A.↗
NASA NTRS1983

An implicit, transonic, full-potential code for cascade flow on H-grid topology

A transonic, full-potential code is developed for computing the flow through two-dimensional cascades using an H-type grid topology that employs an implicit approximate-factorization scheme. The body-conforming H-grid is generated numerically by solving Poisson's equation. The flow-solution algorithm at the coordinate mapping singularity associated with this grid is investigated using two different types of finite-difference schemes. The grid-geometry effect on these schemes is also studied by noting free-stream capturing properties. It is found that by implementing a consistent spatial differencing scheme, the mapping singularities can be resolved numerically, and the grid-geometry-induced error minimized. The code is verified by computing model cascade flow problems.

Kwak, D.↗
NASA NTRS1983

Measurements of the scattering of sound by a line vortex

This paper presents measurements of the phase and magnitude of the scattered field arising from the incidence of a monochromatic plane sound field as a steady vortex. The amplitude of the scattered field was found to vary linearly with the vortex strength, and with the incident wave amplitude and frequency as predicted by solutions based on the Born approximation. The scattered field was observed to be nonsingular in the incidence direction, and this was similar to predictions by the Parabolic Equation Method (PEM) rather than the Born approximation, which predicts singular behavior in the incidence direction.

Horne, W. C.↗
NASA NTRS1987

The quadrupole gravitational lens

An asymmetric lens made of a spherically symmetric galaxy and a quadrupole tidal part allows a very simple pictorial examination of the multiplicity and positions of the images, for any mass density profile. The lens equations can be reduced to one equation in one variable. In the special case of a singular isothermal sphere with a tide (SIST), the lens properties are described by five parameters. Two parameters determine the angular scale and orientation, while three determine the configuration of the images. A comprehensive investigation of the SIST lens is performed, including the variation of the configurations with the parameter values, the critical surfaces in the parameter space, and the extremal solutions. Analytic expressions are derived for the SIST parameters, and additional image positions, in the case when the intensity ratio and positions relative to the galaxy for two of the images are given as input. Quadrupole lenses with a point mass and a pseudoisothermal sphere as the main part are also described in terms of direction diagrams. A direct relation between deflection profile and rotational velocity is derived.

Kovner, Israel↗
NASA NTRS1998

Optimization of Supersonic Transport Trajectories

This paper develops a near-optimal guidance law for generating minimum fuel, time, or cost fixed-range trajectories for supersonic transport aircraft. The approach uses a choice of new state variables along with singular perturbation techniques to time-scale decouple the dynamic equations into multiple equations of single order (second order for the fast dynamics). Application of the maximum principle to each of the decoupled equations, as opposed to application to the original coupled equations, avoids the two point boundary value problem and transforms the problem from one of a functional optimization to one of multiple function optimizations. It is shown that such an approach produces well known aircraft performance results such as minimizing the Brequet factor for minimum fuel consumption and the energy climb path. Furthermore, the new state variables produce a consistent calculation of flight path angle along the trajectory, eliminating one of the deficiencies in the traditional energy state approximation. In addition, jumps in the energy climb path are smoothed out by integration of the original dynamic equations at constant load factor. Numerical results performed for a supersonic transport design show that a pushover dive followed by a pullout at nominal load factors are sufficient maneuvers to smooth the jump.

Ardema, Mark D.↗
NASA NTRS1998

Development of Advanced Methods of Structural and Trajectory Analysis for Transport Aircraft

This paper develops a near-optimal guidance law for generating minimum fuel, time, or cost fixed-range trajectories for supersonic transport aircraft. The approach uses a choice of new state variables along with singular perturbation techniques to time-scale decouple the dynamic equations into multiple equations of single order (second order for the fast dynamics). Application of the maximum principle to each of the decoupled equations, as opposed to application to the original coupled equations, avoids the two point boundary value problem and transforms the problem from one of a functional optimization to one of multiple function optimizations. It is shown that such an approach produces well known aircraft performance results such as minimizing the Brequet factor for minimum fuel consumption and the energy climb path. Furthermore, the new state variables produce a consistent calculation of flight path angle along the trajectory, eliminating one of the deficiencies in the traditional energy state approximation. In addition, jumps in the energy climb path are smoothed out by integration of the original dynamic equations at constant load factor. Numerical results performed for a supersonic transport design show that a pushover dive followed by a pullout at nominal load factors are sufficient maneuvers to smooth the jump.

Ardema, Mark D.↗
NASA NTRS1999

Weakly Nonlinear Description of Parametric Instabilities in Vibrating Flows

This project focuses on the effects of weak dissipation on vibrational flows in microgravity and in particular on (a) the generation of mean flows through viscous effects and their reaction on the flows themselves, and (b) the effects of finite group velocity and dispersion on the resulting dynamics in large domains. The basic mechanism responsible for the generation of such flows is nonlinear and was identified by Schlichting [21] and Longuet-Higgins. However, only recently has it become possible to describe such flows self-consistently in terms of amplitude equations for the parametrically excited waves coupled to a mean flow equation. The derivation of these equations is nontrivial because the limit of zero viscosity is singular. This project focuses on various aspects of this singular problem (i.e., the limit C equivalent to (nu)((g)(h(exp 3)))exp -1/2 << 1,where nu is the kinematic viscosity and h is the liquid depth) in the weakly nonlinear regime. A number of distinct cases is identified depending on the values of the Bond number, the size of the nonlinear terms, distance above threshold and the length scales of interest. The theory provides a quantitative explanation of a number of experiments on the vibration modes of liquid bridges and related experiments on parametric excitation of capillary waves in containers of both small and large aspect ratio. The following is a summary of results obtained thus far.

Knobloch, E.↗