Researchers started their studies on the development and application of computational methods for compressible flows. Particular attention was given to proper numerical treatment of sharp layers occurring in such problems and to general mesh generation capabilities for intricate computational geometries. Mainly finite element methods enhanced with several state-of-the art techniques (such as the streamline-upwind/Petrov-Galerkin, discontinuity capturing, adaptive implicit-explicit, and trouped element-by-element approximate factorization schemes) were employed

Tezduyar, Tayfun E.

English

NASA Technical Reports Server

N90-24996
FINITE ELEMENT FORMULATIONS FOR COMPRESSIBLE FLOWS
Final Report
NASMASEE Summer Faculty Fellowship Program 1989
Johnson Space Center
Prepared by:
Academic Rank:
University and Department:
Tayfun E. Tezduyar
Associate Professor
University of Minnesota
Aerospace Engineering and Mechanics
Minneapolis, Minnesota 55455
NASA/JSC
Directorate:
Division:
Branch:
JSC Colleague:
Date Submitted:
Contract Number:
Engineering
Advanced Programs
Aerosciences
Chien Li, Ph.D.
August 14, 1989
NGT 44-001-800
23-1
https://ntrs.nasa.gov/search.jsp?R=19900015680 2020-03-19T22:00:05+00:00Z
ABSTRACT
We havestartedour studiesrelatedto thedevelopmentandapplicationof computational
methodsfor compressibleflows. Particularattentionis givento propernumericaltreatment
of sharplayersoccuringin suchproblemsandto generalmeshgenerationcapabilitiesfor
intricatecomputationalgeometries.Mainly finite elementmethodsenhancedwith several
state-of-the-arttechniques(suchasthestreamline-upwind/Petrov-Galerldn,discontinuity
capturing, adaptive implicit-explicit, and grouped element-by-elementapproximate
factorizationschemes)will beemployed.
.23-2
INTRODUCTION
The physics and dynamics of the problems involving compressible flows in aerospace
applications are not yet fully understood. Particularly, better understanding of dynamical,
thermal and chemical aspects of the renetry conditions is needed. Prediction of
aerodynamic and heating loads is very important for designing space vehicles. The
conditions such as altitude and speed under which such maneuvering is to be performed
makes the simulation by ground test facilities impossible Therefore, computational
predictions based on the solution of appropriate governing equations become very
important and imperative.
The main objective of this project is to investigate the development and application
of computational tools for aerospace problems with special reference to
compressible flows. These tools need to be not only accurate and reliable but also
efficient and easy to apply to complicated geometries. Unrestricted geometric generality
of a numerical analysis tool will become more and more important and jndispensible
as research in this area targets more and more flow simulations for computational
geometries which give realistic representations of the actual configurations. To this end
we plan to base our efforts on the finite element method (FEM) which is widely
acknowledged to be the most geometrically flexible computational fluid dynamics (CFD)
tool available.
Among the difficulties associated with the numerical simulation of compressible
flows is the treatment of shocks and sharp layers. For hypersonic flows the magnitude of
such sharp variations in the flow field becomes very large and this makes the computation
even more challenging. We plan to use streamline-upwind/Petrov-Galerkin (SUPG)
formulations which are well-known to result in stable and accurate results for
convection-dominated problems in the presence of discontinuites and sharp layers at
moderately high wave numbers. The SUPG method essentially introduces a disspative
effect in the direction of the flow (or in the direction of the characteristic vector).
For very strog discontinuities and sharp layers (i.e. very high wave numbers) we plan
to use the discontinuity capturing (DC) method which introduces a dissipative effect
in the direction of the vector normal to the discontinuity front. This approach produced
very satisfactory results for the model problems studied. Furthermore, with our adaptive
implicit-explicit (AIE) method we will be able to have an implicit refinement around sharp
layers. This will not only increase the efficiency of the computations substantially, but
also, and the most importantly, will result in stable and accurate results near sharp
layers.
23-3
PROBLEM STATEMENT
Let f2 be an open region _md, where nsd is the number of space dimensions. The
boundary of f_ is denoted by F. Spatial coordinates are denoted by x _ f2.
Assuming real gas behavior with equilibrium chemistry, mass, momentum and energy
balance equations are given as follows:
8p
-- + V * ( u p ) = 0 ( mass balance) (1)
8t
m(pu) +V.(upu- o)=0
8t
(momentum balance) (2)
8
--(pe) +Vo(upe- ou +q) = 0
8t
( energy balance ) (3)
where p, u, c, q and e are the density, velocity vector, stress tensor, heat flux vector and
total specific energy, respectively.
Assuming Newtonian fluids which obey the Fourier's law of heat conduction we can
express the constitutive relations as follows:
o = -p I + x (4)
x = Z,(V-u)I+ _t(Vu+(Vu)T) (5)
q = - _cV T (6)
In the above expressions p is the pressure, '_ is the viscous stress tensor, X and I.t are the
two coefficients of viscosity, T is the temperature, and _¢is the conductivity.
23-4
Thetotalenergyecanbewrittenas
e = i + (1/2) Null 2 (7)
where i is the specific internal energy. We assume that for real gases, pressure and
temperature data is available in a form which can be expressed as follows:
p=p(p,i) (8)
T =T(p,i) (9)
Remark
In a more compact vector form, equations(l) - (3) can be written as follows:
_U
-- +V.F(U) = 0
Ot
(10)
where U(x,t) is the vector of unknown variables and F(U) is the flux vector.
We assume that associated with (10) the following boundary and initial conditions are
_iveri:
B U = G on F (11)
U(x,0) = U0(x) on fl (12)
where B (U) is a boundary operator and G and Uo(x) are given functions.
23-5
THE FINITE ELEMENT FORMULATION
Consider a discrctization of _ into clement subdomains G e, e = 1,2 ..... nei where nel is
the number of elements. Let 1"_denote the boundary of f_. Wc assume
nel _e
e=l (13)
nel
ID = n _e (14)
Throughout, we shall assume that trial solutions, U, satisfy BU = G on F and weighting
functions, W, satisfy BW = 0 on r'.
The weak(variational) form of (10) -(12) is given as follows:
W * (_U/bt + V • F ) dfl = 0 (15)
f W * (U(x,O)-Uo(x)) df_ = 0 (16)
Remark
If the weighting functions are selected from the same set of functions as the trial solutions
then (15) is a regular (Bubnov-) Galerkin formulation; else, it is a Petrov-GalerkiB
formulation. The significance of the Petrov-Galerkin formulation will be explained in the
sequel.
Spatial discretization of (15-16) is carried out by expanding U and W in terms of a set of
finite element basis functions corresponding to (13-14). For example the expression for U
might take the form ....
23-6
U(x,t) =_ NB(X) Us(t) (17)
B
where B is the nodal index, N B is the finite element basis function associated with node B,
and U a is the nodal value of U.
The spatial discretization leads to a set of semi-discrete, ordinary differential equations:
M V + N (V) = F (18)
with the initial condition:
V(0) = V0 (19)
where V is the vector of nodal values of U, V 0 is its given initial value, M is a generalized
"inertia" matrix, N is a nonlinear vector function of V, and F is a "force" vector. A
superposed dot denotes time differentiation.
Given the initial condition (19), equation (18) can be integrated by an appropriate
temporal discretization scheme.
23-7
tCONCLUSIONS
We have started with our development efforts and preliminary computations for finite
element formulations applied to compressible flows.We are particularly interested in the
proper numerical treatment of sharp layers occuring in such problems We also target the
most general mesh generation capabilities for intricate computational geometries. Our
methods will be enhanced with several state-of-the-art techniques such as the streamline-
upwind/Petrov-Galerkin, discontinuity capturing, adaptive implicit-explicit, and grouped
element-by-element approximate factodzation schemes. Our preliminary computations
show that these methods are stable, have minimal numerical dissipation, and are very
reliable.
23-8


Finite element formulations for compressible flows

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