A second-order projection method for the time-dependent, incompressible Navier Stokes equations i... more A second-order projection method for the time-dependent, incompressible Navier Stokes equations is presented. The numerical method consists of three separate parts: a time-stepping strategy that ensures second-order accuracy, a numerical approximation of the Hodge projection, and a second-order discretization of the nonlinear convection terms. The basic requirements for second-order temporal accuracy are discussed in a semidiscrete form. Approximation of the projection using a discrete Galerkin formulation based on determination of a local basis for discretely divergence-free vector fields is described. A specialized, second-order Godunov procedure is developed for discretization of the nonlinear convective terms. The Godunov method remains stable and non-oscillatory as Reynolds number tends to infinity. From a linear algebraic point of view, the overall algorithm requires only the solution of symmetric, positive-definite systems that are ideally suited to iterative methods. Numerical results are presented validating the convergence properties of the method.
22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Jun 24, 1991
The dynamics of unconfined, spatially developing shear layers is studied by numerical solutions o... more The dynamics of unconfined, spatially developing shear layers is studied by numerical solutions of the time-dependent Euler equations using a second-order Godunov scheme. Effects of density and velocity variations between the two streams of the shear layer are studied and color graphics is used to show more clearly the entrainment process of the surrounding streams. The calculations demonstrated that the evolution of the mean flow was dominated by two-dimensional, inviscid effect.s. The I.m.s. fluctuating velocity and density profiles were found to be in good agreement with the measurements of Oster and \\'ygnanski and of Konrad, except for the peak value of the v' profile.
In this paper we describe a second-order projection method for the time-dependent, incompressible... more In this paper we describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected ...
We present here computational results using second order Godunov methods for timedependent Euleri... more We present here computational results using second order Godunov methods for timedependent Eulerian gas dynamics with a general (convex) equation of state. The algorithm rsed in most of the calculations is described in detail in [3). An unsplit scheme was used in conj ac tion with local adaptive mesh refinement for the results in Figure 3; see [1] and the referc is cited there for the construction of this version of the scheme. For the well-known problem of planar shock wave diffraction by a wedge, a direct comparison! of computational results and the experimental record has been completed for several differet cases, [5|. Reproduced in Figure 1 are the results for a shock wave Mich number M,-2.0S and wedge angle 9-27*. The experimantal flowfield for this case of single Mach reflection is in equilibrium and viscous effects are localized at the wedge comer and the content surfaceboundary layer interaction. The infinite-fringe interferogram is shown in Fig. la, the numerical results with the same isopycnics (i.e., constant density lines) in Fig. lb, and the wall density values are compared in Fig. lc. It is seen that excellent agreement is obtained in all respects.
DISCI.AIMER Thi~ document luas prepared .~ an account of work sponsored b)' an .Keft('~' of the U... more DISCI.AIMER Thi~ document luas prepared .~ an account of work sponsored b)' an .Keft('~' of the Uniled Slates Go'ternment. Neither the United States Gonrnment ~r the Uni\'ersit), of C.morai. nor an)' of their employees., m.akes aay "arnnly. npress or implied. or assumes .n~' le.allillbilit) or responsibility for the attUrae). completeness.. or asefulness of an) inform.alion. apparatus. produd. or p~ diSC'losed. or reprewnts that its usc "'ol.lld not infringe privattl}' O'I"ned rights. Re(ereMe herein ' 0 an)' specific commerdal products. process. or service b) trade name. trademark. manuladurer. or other"'ise. does not necessaril), can.titute or imply Us t'ndorseme-nt. recommendation. or fa\lorina b~' the United States Government or tbe Universit)' of California. The "jews and opinions or authors npressed herein do not neces.saril,' stale or renect those of the United States Governmenl or the Universit) of California. and shaU not be used for ad\ertisina or prodoct endorsement purposes.. .
19th AIAA, Fluid Dynamics, Plasma Dynamics, and Lasers Conference, 1987
The Eulerian second-order Godunoy scheme is extended to treat a mixture of nonequ1ltbrium. chemic... more The Eulerian second-order Godunoy scheme is extended to treat a mixture of nonequ1ltbrium. chemically reacting gases. We consider only the case of high temperature air here, although the scheme ;s more generally applicable. Several planar oblique shock waye calculations are discussed, including direct comparison with experimental data. The new results are an improvement over our previous gas dynamics calculations for the same problems.
This report was prepared as an account of work sponsored by the United States Government. Neither... more This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information,' apparatus, product or process disclosed; or represents that its use would not infringe privately owned rights.
Numerical simulations are described for three idealized shear layer problems typical of shock ref... more Numerical simulations are described for three idealized shear layer problems typical of shock reflection flow fields: free shear layers (corresponding to a slip line emanating from a triple point), a wall shear layer (approximating a boundary layer behind a shock), and a wall jet (similar to those found in precursor and double-Mach-stem flows). The dynamic evolution of the shear layers was followed by means of numerical solutions of the inviscid conservation laws of gasdynamics. The inviscid shear layer discontinuity was de-singularized by initializing the flowfield with a Tanh(y) velocity profile that, in effect, allows one to resolve the shear layer on the computational mesh. The inflow velocity profile was then perturbed with the fundamental frequency (from linear stability analysis) and its subharmonics. The macroscopic features of the calculations (e.g., the formation and growth of large-scale rotational structures, the visual spreading rates, and the mean flow profiles) agree ...
A hybrid implicit-explicit scheme is developed for Eulerian hydrodynamics. The hybridi~atlon is a... more A hybrid implicit-explicit scheme is developed for Eulerian hydrodynamics. The hybridi~atlon is a continuous switch and operates on each characteristic field separately. The explicit scheme is a version of the second-order Godunov scheme; the implicit method is only firstorder accurate in time but leads to Ii block tridiagonal matrix inversion for efficiency and is unconditionally stable for the case of linear advection. The methodology is described for the cases of linear advection, for nonlinear scalar problems, and for gas dynamics. An important element of our work is the use of a modified Engquist-osher flux function In place of the Godunov flux. Several numerical results are presented to demonstrate the properties of the method, especially stable numerical shocks at very high CFL numbers and second-order accurate steady states. Cl 1995 Academic Press. Inc. * The support ofthe Defense Nuclear Agency, MIPR 90-567 is gratefully acknowledged. t The support of the Applied Mathematical Sciences Program of the u.s.
A second-order projection method for the time-dependent, incompressible Navier Stokes equations i... more A second-order projection method for the time-dependent, incompressible Navier Stokes equations is presented. The numerical method consists of three separate parts: a time-stepping strategy that ensures second-order accuracy, a numerical approximation of the Hodge projection, and a second-order discretization of the nonlinear convection terms. The basic requirements for second-order temporal accuracy are discussed in a semidiscrete form. Approximation of the projection using a discrete Galerkin formulation based on determination of a local basis for discretely divergence-free vector fields is described. A specialized, second-order Godunov procedure is developed for discretization of the nonlinear convective terms. The Godunov method remains stable and non-oscillatory as Reynolds number tends to infinity. From a linear algebraic point of view, the overall algorithm requires only the solution of symmetric, positive-definite systems that are ideally suited to iterative methods. Numerical results are presented validating the convergence properties of the method.
22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Jun 24, 1991
The dynamics of unconfined, spatially developing shear layers is studied by numerical solutions o... more The dynamics of unconfined, spatially developing shear layers is studied by numerical solutions of the time-dependent Euler equations using a second-order Godunov scheme. Effects of density and velocity variations between the two streams of the shear layer are studied and color graphics is used to show more clearly the entrainment process of the surrounding streams. The calculations demonstrated that the evolution of the mean flow was dominated by two-dimensional, inviscid effect.s. The I.m.s. fluctuating velocity and density profiles were found to be in good agreement with the measurements of Oster and \\'ygnanski and of Konrad, except for the peak value of the v' profile.
In this paper we describe a second-order projection method for the time-dependent, incompressible... more In this paper we describe a second-order projection method for the time-dependent, incompressible Navier-Stokes equations. As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict intermediate velocities which are then projected ...
We present here computational results using second order Godunov methods for timedependent Euleri... more We present here computational results using second order Godunov methods for timedependent Eulerian gas dynamics with a general (convex) equation of state. The algorithm rsed in most of the calculations is described in detail in [3). An unsplit scheme was used in conj ac tion with local adaptive mesh refinement for the results in Figure 3; see [1] and the referc is cited there for the construction of this version of the scheme. For the well-known problem of planar shock wave diffraction by a wedge, a direct comparison! of computational results and the experimental record has been completed for several differet cases, [5|. Reproduced in Figure 1 are the results for a shock wave Mich number M,-2.0S and wedge angle 9-27*. The experimantal flowfield for this case of single Mach reflection is in equilibrium and viscous effects are localized at the wedge comer and the content surfaceboundary layer interaction. The infinite-fringe interferogram is shown in Fig. la, the numerical results with the same isopycnics (i.e., constant density lines) in Fig. lb, and the wall density values are compared in Fig. lc. It is seen that excellent agreement is obtained in all respects.
DISCI.AIMER Thi~ document luas prepared .~ an account of work sponsored b)' an .Keft('~' of the U... more DISCI.AIMER Thi~ document luas prepared .~ an account of work sponsored b)' an .Keft('~' of the Uniled Slates Go'ternment. Neither the United States Gonrnment ~r the Uni\'ersit), of C.morai. nor an)' of their employees., m.akes aay "arnnly. npress or implied. or assumes .n~' le.allillbilit) or responsibility for the attUrae). completeness.. or asefulness of an) inform.alion. apparatus. produd. or p~ diSC'losed. or reprewnts that its usc "'ol.lld not infringe privattl}' O'I"ned rights. Re(ereMe herein ' 0 an)' specific commerdal products. process. or service b) trade name. trademark. manuladurer. or other"'ise. does not necessaril), can.titute or imply Us t'ndorseme-nt. recommendation. or fa\lorina b~' the United States Government or tbe Universit)' of California. The "jews and opinions or authors npressed herein do not neces.saril,' stale or renect those of the United States Governmenl or the Universit) of California. and shaU not be used for ad\ertisina or prodoct endorsement purposes.. .
19th AIAA, Fluid Dynamics, Plasma Dynamics, and Lasers Conference, 1987
The Eulerian second-order Godunoy scheme is extended to treat a mixture of nonequ1ltbrium. chemic... more The Eulerian second-order Godunoy scheme is extended to treat a mixture of nonequ1ltbrium. chemically reacting gases. We consider only the case of high temperature air here, although the scheme ;s more generally applicable. Several planar oblique shock waye calculations are discussed, including direct comparison with experimental data. The new results are an improvement over our previous gas dynamics calculations for the same problems.
This report was prepared as an account of work sponsored by the United States Government. Neither... more This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information,' apparatus, product or process disclosed; or represents that its use would not infringe privately owned rights.
Numerical simulations are described for three idealized shear layer problems typical of shock ref... more Numerical simulations are described for three idealized shear layer problems typical of shock reflection flow fields: free shear layers (corresponding to a slip line emanating from a triple point), a wall shear layer (approximating a boundary layer behind a shock), and a wall jet (similar to those found in precursor and double-Mach-stem flows). The dynamic evolution of the shear layers was followed by means of numerical solutions of the inviscid conservation laws of gasdynamics. The inviscid shear layer discontinuity was de-singularized by initializing the flowfield with a Tanh(y) velocity profile that, in effect, allows one to resolve the shear layer on the computational mesh. The inflow velocity profile was then perturbed with the fundamental frequency (from linear stability analysis) and its subharmonics. The macroscopic features of the calculations (e.g., the formation and growth of large-scale rotational structures, the visual spreading rates, and the mean flow profiles) agree ...
A hybrid implicit-explicit scheme is developed for Eulerian hydrodynamics. The hybridi~atlon is a... more A hybrid implicit-explicit scheme is developed for Eulerian hydrodynamics. The hybridi~atlon is a continuous switch and operates on each characteristic field separately. The explicit scheme is a version of the second-order Godunov scheme; the implicit method is only firstorder accurate in time but leads to Ii block tridiagonal matrix inversion for efficiency and is unconditionally stable for the case of linear advection. The methodology is described for the cases of linear advection, for nonlinear scalar problems, and for gas dynamics. An important element of our work is the use of a modified Engquist-osher flux function In place of the Godunov flux. Several numerical results are presented to demonstrate the properties of the method, especially stable numerical shocks at very high CFL numbers and second-order accurate steady states. Cl 1995 Academic Press. Inc. * The support ofthe Defense Nuclear Agency, MIPR 90-567 is gratefully acknowledged. t The support of the Applied Mathematical Sciences Program of the u.s.
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Papers by Harland Glaz