AD-A171 813DNA-TR-86-92

CONSERVATION EQUATIONS FOR A NONSTEADY FLOW OFA COMPRESSIBLE VISCOUS SINGLE-PHASE FLUID INVARIOUS COORDINATE SYSTEMS

Gordon C. K. YehAllen L. KuhlR & D AssociatesP. O, Box 9695Marina del Rey, CA 90295-2095

31 December 1984

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Ii. TingL IIftowi S 3sm"IW OmafrtnICONSERVATION EQUATIONS FOR A NONSTEADY FLOW OF A COMPRESSIBLE VISCOUSSINGLE-PHASE FLUID IN VARIOUS COORDIIIATE SYSTEMS12. PERSONA141L AUTMMS

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F1.1LO IGROU IUE4RQU_ Nonsteady Flow20 1 -Compressible Viscous Fluid

1-2-0. -1.- Momentum Conservation Equation""ASTRACT (Ca W an 9M it &WOW W 6%d ,,, 6 ,,oCk ,,

Based on a generalization of the Reynolds transport theorem the mass,momentum and energy conservation equations for a nonsteady flow of a com-pressible viscous single-phase fluid (expressed in vector and dyadicnotations) are first formulated in integral form (in terms of movingvolume and surface elements), next converted into differential form andthen transformed in terms of orthogonal curvilinear coordinates. Special-ized equations are obtained for three-dinmensional flows in Cartesian,cylindrical and spherical coordinates. These equations can then bereduced to corresponding equations for one4 and twoý'dimensional flows invarious coordinates. Equations for the vorticity, entropy and enthalpyand Bernoulli equation are also summarized.

2M OISTATIUTIONIAVAILANILITY GP ABSTRACT 21 ABSTRACT SECURI1TV CLASSIFICATIONOUNCI.ASSMJFIUNLIMITO SSAMe AS ,AP. DOTIC UtERS UNCLASSIFIED

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18. SUBJECT TERMS (Continued)

Energy Conservation EquationCartesian, Cylindrical, Spherical, and Orthogonal Curvilinear CoordinatesOne, Two, and Three Dimensional FlowsVector and Dyadic (Second Order Tensor) NotationsBernoulli EquationVorticityEntropyEnthalpyKinematic Transport TheoremIntegral and Differential FormulationsConservative and Nonconservative Forms

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UNCLASSI FIEL,

TABLE OF CONTENTS

Section Page

1 INTRODUCTION 1

2 INTEGRAL FORMULATION 2

2.1 Kinematic Transport Theorem 2

2.2 Conservation Laws 5

2.3 Constitutive Relations 7

2.4 Thermodynamic/Transport Properties 8

3 CONSERVATION LAWS IN DIFFERENTIAL FORM 11

3.1 Conversion by Generalized Definition

of Operator V 11

3.2 Conservative Form 12

3.3 Nonconservative Form 133.4 Vorticity Distribution 15

3.! Bernoulli Equation 20

4 TRANSFORMATION OF CONSERVATION EQUATIONS INTERMS OF ORTHOGONAL CURVILINEAR COORDINATES 24

5 CONSERVATION EQUATIONS IN CARTESIANCOORDINATES 27

6 CONSERVATION EQUATIONS IN CYLINDRICALCOORDINATES 28

7 CONSERVATION EQUATIONS IN SPHERICALCOORDINATES 30

8 COMPACT ONE-DIMENSIONAL FORMULATION 32

9 CONCLUSIONS 33

10 LIST OF REFERENCES 34

iii

SECTION 1

INTRODUCTION

The governing equations in fluid mechanics are usually

expressed in terms of Cartesian coordinates. They are notreadily convertible into forms useful for practical appli-cations to one-, two- and three-dimensional cylindrical and

spherical problems. One of the purposes of this report is totransform the mass, momentum and energy conservation equationsfor a nonsteady flow of a compressible viscous single-phasefluid from expressions in vector and dyadic notations to those

in terms of orthogonal cuvilinear coordinates and then tospecialize the equations for Cartesian, cylindrical andspherical coordinates.

Another purpose of this report is to provide a theoretical

basis for usage of Eulerian sliding grids in modern computa-tional fluid mechanics by establishing a kinematic transporttheorem which is a generalization of the Reynolds transport

theorem. Based upon the theorem the conservation equationsin integral form are first formulated in terms of movingvolume and surface elements and then converted into the corres-ponding equations in differential form by using the general-ized definition of the vector operator V.

In order to present an overall survey of the basic equationsthis rt~p.,rt also summarizes the equations for the vorticity,

entropw entha.py and Bernoulli equation.

SECTION 2

INTEGRAL FORMULATION

2.1 KINEMATIC TRANSPORT THEOREM.

Let . = (x1 , x 2 , x3 ) denote the rectangular spatial ("Eulerian")

coordinates which identify a fixed point in space. Let X =(X1 , X2, X3 ) denote the rectangular material ("Lagrangian")coordinates which identify a fluid particle in motion. LetF = F(x, t) represent any arbitrary single-valued scalar orvector point function (of position x and time t) possessingcontinuous derivatives. The function

S= • ~F~x_, t•d f • Ft_ (X_, t),t]J o, )V o

where V = V(t) denotes a material volume (that is, a volumemoving with the fluid), is a well-defined function of time.In Eq. (1) the Jacobian (Ref. 1, p.33)

dV i}J # X2 x3 ) axJ 0,v~ det(-_') (i 1 ,203: aa 1,203) (2)3V0 aocXl 2 X3)

relates the element dV of the moving volume V in the x-variables

to the element dV0 of the fixed volume Vo (o) 0 V(t) as t , 0in the X-variables.

Using Euler's expansion formula (Ref. 2)

a- = JV'V (3)

2

(where v is the velocity vector of the fluid motion) and the

relation between the material time derivative and the spatial

derivatives (Ref. 3, Eq. (3.6))

d aa+ V(4

we can express the material time derivative of 'I as

f [F + FOv.vj JdVQ

af[ ~+ VOW+F(V*V)JdV

uf [•+divcvv y .v5 )

Then in view of Green's transformation (Ref. 1, Sq. (7.2))

for any vector or tensor field *

.0 div *V dý (dA ndA)()

we obtain

3

f FVnd + AMA (7)

or

f Fd -, Fv-ndA +,* (7a)

where V denotes the volume fixed in space which instantaneouslycoincides with the material volume V, A denotes the surfacebounding the volume V and n denotes the unit vector along theoutward normal to A. Eq. (7a) is known as the transport

theorem of Reynolds (Ref. 1, Eq. (25.4) and Ref. 4, § 14).It is a kinematic relation independent of any meaning attached

dto F. All fluid physics is contained in the term.

Now let us consider a volume V a V(t) with bounding surface

A - A(t) sliding with respect to the fluid and obeying therelation

WdV) (- _ (8)

where u is the local surface velocity. We can write

FaE(dr) - Fu-sdA ( 9)V

To each time instant t when the fixed volume V coincides withthe material volume V there corresponds a time instant t1 whenthe m~oving volume V coincides with the same material volumeSinstantaneously. At t, Eq. (9) becomes

Ir ~dV) Pmf (10)

4

where F1 = F(x,t 1 ), u 1 = u(x,tI) and -l(dV) = f5(dV)t = tI"

Since V, A and n in Eq. (10) have the same meaning as thosein Eq. (7) and dV and dA (at t) in Eq. (7) equal to dý and

dW (at t1) in Eq. (10) we may add Eqs. (7) and (10) to obtain

ý A F=- (Fv_ FU1) .ndA + t • (11)

In case V and V can coincide with V at the same time (tI = t)

(for example, if u1 =,0 or u! = v) Eq. (11) becomes

S d4 5-(FdV) F - F(V " u).ndA + mv U(lia)

or (since V is fixed in space)

dVn (lib)P 1A F (v u) +jM

This is a generalization of the Reynolds transport theoremand gives the fundamental transport formula for the time vari-ation of the volume integral of any fluid state quantity Fover a moving volume which instantaneously coincides withthe material volume. As u = 0 (V - V, hence tI = t) Eq. (llb)

reduces to Eq. (7a) which is the basis of the Eulerian gridfto

representation in numerical algorithm. As U -- V (V = V, hencetI f t) Eq. (llb) becomes a statement of the Lagrangian mesh

representation. Ref. 5 shows an example of the application

of Eq. (llb).

2.2 CONSERVATION LAWS.

The mass, momentum and energy conservation equations for a non-

steady flow of a compressible, viscous, heat-conducting fluid

5

derived and discussed in reference books on gas dynamics (see,

for example, Refs. 6 to 13) have been summarized in recent

books on computational fluid mechanics (see, for example,

Refs. 14 to 16) in compact vector and dyadic (second order

tensor) notations. In a fixed Eulerian frame of reference

these equations in integral form (which is more basic thanthe differential form in expressing the physical laws) are asfollows:

mass: a PA pv.dA = 0 (12)

momentum: -f pvdV + pv(vgn)dA g'ndA +fjdV , (13)

energy: -fpEdV + pEv.ndA = fA ) - a)ndA

"+ f dV (14)

In these equations, p is the density, E the total specific

energy ("specific" means per unit mass)

E e (15)

(where e is the specific internal energy), g is the stress

tensor, a the heat-flux vector, f the external force vector

per unit volume, and G is the energy generation per unit

volume, and V is a control volume fixed in space, A is the

-, surface bounding V, and n is the unit vector along the out-

-. ward normal to A.

6

• ..•• .".• ' V.• .' • • • -,•. •'•... '.•••.'•g.N'. •. •''• '' ".".•.". .. •"

In these equations the properties of the fluid need not be

continuous functions of space and time.

By comparison of Eq. (7a) with Eqs. (12) to (14) the function

F may be identified as p, pv and pE respectively and the term

dM-/dt as the right-hand side terms of each of these equations.

Then in terms of the moving elements (dV = dV and dA = dA

instantaneously) the mass, momentum and energy conservation

equations in case tI t (exactly or approximately) can be

expressed exactly or approximately as a single vector equation

according to Eq. (llb):

f dV - - W_ - u)' ndA

+ fA _ZndA IQV (16)

where

P 0Z£ 0

Eq. (16) provides a theoretical basis for usage of Eulerian

sliding grids in finite-difference numerical algorithm (see,

for example, Ref. 5).

2.3 CONSTITUTIVE IRELATIONS.

The basic dependent variables in Eqs. (12) to (14) or Eqs. (16)

are p, v and E (or e). Constitutive relations for g and q

must be added to these equations in order to obtain a closed

system. We are concerned here with the case of Newtonian

fluids, i.e., by definition, fluids such that the stress tensor

7

I

is a linear functicn of the velocity gradient. From this

definition, excluding the existence of distributed forcecouples, results Newton's law, also called the Navier-Stokes

law, for 4:

with

X= (V.v)_ + 2UR (19)

and

S' [jv .+ (vV)tJ (20)

the superscript t denoting the transpose of a tensor. InEqs. (18) to (20) p is the pressure, I is the viscous (ordeviatoric) stress tensor, ; is the unit tensor, U and X are

the first (shear) and second (dilatational) coefficients ofviscosity, and 2 is the tensor of rates of deformation.Furthermore, the fluid is assumed to obey Fourier's law ofheat conduction for %t

a=-kVT , (21)

where T is the absolute temperature. and k is the thermal con-

ductivity coefficient. Many fluids, in particular air andwater, follow Newton's law and Fourier's law.

22.4 THEROMDYN•AHIC/TRANSPORT PROPERTIES.

The state variables 0, e, p, T and the specific entropy S are

connected by thermodynamic relations (assuming local thermo-dynamic equilibrium).

We consider the case of a simple fluid such that all its thermo-dynamic properties can be deduced from a single fundamental

relationship which, for a compressible fluid, can be chosen of

the type

S = S(p,e) • (22)

Fxom tCis zelationship p and T are obtained in terms of the

basic vaiiableL p and e from

p =p 2 T(•Se T = iT . (23)

An important special case is a perfect gas with constant spe-cific heats c and v.* For inuch a gas the laws of state are

p = (y-l)pe, (Y C p/CV) (24)

ande = CvT • (25)

The viscosity and thermal conductivity coefficients dependon the local thermodynamic state; in most conditions they depend

only on the temperature;

ii i i(T), X X(T), k = kT) • (26)

The coefficient

K 3X + 2p (27)

is called bulk viscosity coefficient In the "Stokes relation"(Ref. 3, p. 238\

3X + 2o,= 0 (2C)

0

it is assumed to be zero. However, except for very special

conditions, for example, monatomic gases, there is no reason

to assume 3X = -2p. (Ref. 9, p.337 and Ref. 7, p.540.)

10

SECTION 3

CONSERVATION LAWS IN DIFFERENTIAL FORM

3.1 CONVERSION BY GENERALIZED DEFINITION OF OPERATOR V.

If the properties of the flui•d are continuous and sufficientlydifferentiable in some domain of space and time, then the con-

servation equations in integral form can be converted into anequivalent set of partial differential equations through ageneralized definition of the vector operator V (Ref. 17, p.40)

V.0 = i V an!dA (29)

where 0 is an unspecified (scalar, vector or dyadic) functionof position, V is the volume enclosed by a surface A to whichthe point P at which Vý is to be calculated remains interior,while the largest dimension of A tends to zero, and the multi-plication in Vý may be scalar, vector or dyadic. In particular

when • - b, a vector, the scalar product becomes

Veb=lim bendA (30)V.0.

which is a scalar, and when 4 = •, • dyadic, the dot product

becomes

V.0 V - A'

which is a vector. Eqs. (30)and (31) can be also obtained bydividing Eq. (6) by V and then taking the limits of both sidesas V approaches to zero.

11

3.2 CONSERVATIVE FORM.

Dividing Eqs. (12) to (14) by V, taking the limit of everyterm as V approaches zero and applying Eqs. (30) and (31) we"obtain the conservation equations in differential form:

mas.s: + V(pv) = 0 , (32)

momentum: at (pv) + V°(pvv_- =Y) = f , (33)

energy: a (pE) + V.+(pE.- ._+ ,) , _ G • (34)

An alternate form of the energy equation Eq. (34) is in terms

of the enthalpy per unit mass

hw, e + p/p (35)

Substituting

12E -h + v p/p H -p/p (36)

into Eq. (34) we obtain

S (pH) + V.(p v-pv -- .v_ + ) f.v + + G (37)

-- at

where

=h + Zv + P/P (38)

is the total enthalpy per unit mass (which is also known as the"specific stagnation enthalpy" or for nonviscous and nonheat-conducting fluid in steady motion the *total energy per unitmass*, -fif. 18, p. 33).

12.4

Substituting Eq. (36) into Eq. (14) we obtain the integral

form of the conservation equation for pH in terms of fixedelements dV and dA

SfpdV + fA pHv.ndA

-inJ(g.-X-SL~). dA +f/v(f. +G+ ,)dV . (37a)

The corresponding conservation equation in terms of movingelements dý and d• in case tI - t is Eq. (16) with

wa (pH). _q a (gI-~pvQ (.v+G+ )(17a)

3.3 NONCONSERVATIVL FORM.

Eqs. (32) to (34) are mass, momentum and energy equations in"conservative" or"divergence" form. (See Section I11-A-3 ofRef. 14 for the meaning and beneficial effects of using thisform). The corresponding equations in nonconservative formin which & follows Newton's law are (Ref. 16, Section 1.1):

at (39)mass: r + 0v'v'o , 39

momentum: o d + Vp - f + UV2V + (X+ui)V(V.v)

+(V.v)VX + 29.Vu , (40)

energy: p d + pV-v - O-V.a + G ,(41)

13

where 0 is the dissipation function

O= I:Vv = )(V'v) 2 + 2uD:D (42)

and rt is the material derivative given in Eq. (4).

An alternate forn of the energy equation Eq. (41) is in terms

of the specific entropy S such that (Ref. 3, Eq. (33.1))

TdS = de + pd(I) (43)

By use of Eqs. (39) and (43) we obtain from Eq. (41)

dSPT at- 4- V-q + G . (44)

Adding Eq. (32)(multiplied by S) and Eq. (44) (divided by T)

we have the conservative form of the conservation equationfor PS

(PS) + V. (pSv) = (O-V.a+G) . (44a)

Taking the integral of Eq. (44a) over a fixed control volume Vand applying Green's transformation Eq. (6) we obtain theintegral form of the conservation equation for pS in terms offixed elements dV and dA

4PSdV + fpSv.ndA

S(1-V.4+G.dV (44b)

14

The corresponding conservation equation in terms of movingelements 6 and dk in case tI = t is Eq. (16) with

W_ (pS), t- (0), Q = (O/T-V-S/T+G/T) (1.7b)

3.4 VORTICITY DISTRIBUTION.

The vorticity vector

S- V x v (45)

gives the intrinsic rotation of each fluid element.

By taking the curl of both sides of Eq. (40) (divided by p)and noting that

? * a =0 (46)

and

V x V (scalar function) -0 (47)

we obtain the general equation of vorticity distribution

dil a ~ + Q( -v + V 1 ) x Vpv

V x f + V (1) xC f + Vi V2 - V(RL) x V xP 0 -

+ V(•.)x V(V.v) - _ VX x V(V.v)Pp

+ (V.v)1V(!) x 71 + V x (R.V70 + V(1) x(DVi)4)- oP (=.u (48)

If u and X are not functions of space variables Eq. (48)reduces to

15

d_ (flV)v + 11(V.v) + V( x Vp

-V x f +IV(-xf + 7 2• P

+ (X2p V(*~ 161)(U ) xVxfl (49)

If the flow is incompressible which is characterized by the

condition

V.v - 0 (50)

and implies that p is constant along a fluid particle trajec-

tory but not necessarily independent of space variables (as

in stratified flowspRef. 16, Section 1.3) Eq. (48) reduces to

-V( )xV7_.4 +. IV (9.VU) + V(A) x (R-vu). 5!1),0 P

If, in addition, P is constant everywhere Eq. (51) reduces to

,']V x f÷+ V20l

1- -.v 0 vx_-2 .v).1

4

1

p

If, in addition, p is not a function of space variablesEq. (52) reduces to (Ref. 16, Eq. (1.37))

dl1 2M-_.V)v = V. x f + R V S . (53)

This equation is usually associated with an equation for a

solenoid stream-function vector IF such that

v = V x , (54)

which automatically satisfies the incompressibility condition

Eq. (50). The equation satisfied by T is derived by applyingthe curl operator to Eq. (54) and using definition Eq. (45)

to obtain

V2T + a ,0 (55)

since

V - 0 (56)

as T is solenoid.

Let us return now to consider the case of compressible flow.

For an ideal fluid (*a)w0) either Eq. (48) or Eq. (49) reducesto [Ref. 18, Eq. (7-20)]

df- (silV)v_ + n_(V-V)

V x f + x f_- V Vp) (57)

17

Another form of this equation can be obtained by usingEq. (43)[Ref. 18, Eq. (7-211 *

i- (f_.V)v + f_(V.v)

v x f + 7(1) x f + VT x VS (58)P P

If p is a function of p only the last term of Eq. (57) can

be written as

( Vp) V XV~i'rdA (59)

which is zero according to Eq. (47). If the specific entropyS is a constant, then the last term of Eq, (58) is zerobecause VS B o. 2oth conditions are satisfied if the fluid isisentropic, i.e., the fluid has the same entropy everywhere.In the absence of any external force f, if a is zero for oneinstant at every point of the flow field then da/dt - 0.Therefore the vorticity at every point of the field willremain zero and the flow is irrotational. On the other hand,if the flow is not isentropic, i.e., the fluid has differententropy at different points of the field, the last term ofEq. (58) will cause the vorticity to be different from zero inthe next instant. Therefore, nonisentropy flows cannot beirrotational. Hence irrotationality impliet. isentropy, butisentixopy does not imply irrotationality.

Multiplying Eq. (48) by p, expanding the term (p!I.V)v byformula (X) on p. 44 of Ref. 17 and adding Eq. (32) (multipliedby 0) we have the conservative form of the conservation equa-tion for pa

18

ia

i (P0) + V* (pv_ Q) = -V. (pA V) + 2I (48a)

where

Q_.- -zv(. (,Q)]j - pfl(V.v_) - , (•)xVp, ,. 1••_ •ýV[V-0101 011-v) P(!)X P+ V(!)Xf + VXf

"+U~ + PV(ý±+3±)xV(V.Z) _-VR)xx

"+ o(Iv)V_ )xV), - V.xV(V-v) + 2Vxl(-VU)

2+ pV( 1)x (R.VU) (48b)

Taking the integral of Eq. (48a) over a fixed control volume Vand applying Grean's transformation Eq. (6) and Egs. (6) and(7) on p. 53 of Ref. 17 we obtain the integral form of the

conservation equation for pa in terms of fixed elemnts dVand dA

f - PII.n aA+ dV (48c)

The corresponding conservation equation in terms of moving

elements dý and dA in case ti - t is Eq. (16) with

- (o_ I)), - 1 ) • (17c)

Combining Eqs. (17a), (17b) and (17c) with Eqs. (17) we con-

vert Eq. (16) to a vector equation representing six conser-vation equations in integral form with

19

Pv a / f

W OE' 1 =- '-~ a f'v +G .(17d)

PHIV - S p! .v + G +I at

PS/ 0 O/T - V._q/T + G/T

3.5 BERNOULLI EQUATION.

As discussed in Ref. 18, Sectien A,6, aside from the boundarylayer, the effects of viscosity and heat conduction can beneglected for the majority of gas dynamics problems. Further-more, the heat addition, e,.-ept in problems involving combus-tion, is either zero or r)ery small. Then under ordinaryconditions the gas behaves very much like an ideal fluid(i.e., a nonviscous and nonheat-conducting fluid). Therefore,one of the fundamaental problems of gas dynamics is to studythe adiabatic flow of an ideal gas.

Let us further assume that the density of the fluid is a func-tion of pressure only (i.e., the fluid is barotropic, Ref. 3,p. 150) and that the external force per unit mass is repre-sentable by a potentialýP

f/p =(60)

Then by using the relation (Ref. 17, p. 44)1 2

(v.V)v = L Vv2 - v x (V x v) (61)

and definition of _ (Eq. (45))Eq. (40) can be written as

20

-V x vVP - V( v) V - -vx (62)

where

X + v2 + (p (63)

Let s be a parameter along an arbitrary curve in space. Atany point (with position vector r) along this curve dr/ds isa unit tangent vector to the curve. The component of Eq. (C 2)

in the direction of dr/ds is

av dr dr dr• ••-(v_ X e) * •• •VX (64)

Since

dr 8V/.v dr\d av dr

we may express Eq. (64) as (Ref.19, p. 11)

dr dr . VX (66)

where

X , 2 ½ ,, + ,, + fd (67)

At any point in time and at every point in the flow fieldlet dr/ds represent a tangent vector to a streamline (in thedirection of v) or a vortex line (in the direction of 0).

21

Is L

Then the scalar triple product on the left side of Eq. (66)

is zero and the Bernoulli's equation

X = a constant (68)

is valid along any streamline or vortex line but may have

different constants for different streamlines and vortex lines.

For an irrotational flow a = 0 and there is a velocity poten-

tial *:

v -Vt (69)

Eq. (68) is valid along every curve in the flow field sincein Eq. (66) VX = 0 everywhere. Furthermore, from Eqs. (62)

and (69) we obtain the Bernoulli's equation for a potential

flow

f .+ v2 + ( + -a constant, (701

which holds throughout the flow field. The integral in Eq. (70)must be computed with isentropic pressure-density relationbecause as discussed in the previous section, irrotationalmotion can generally be maintained only by constant specific

entropy throughout the field.

For a perfect gas with constant specific heats, the equation

of state can be written in terms of S, p, p as (Ref. 18,Eq. (7-16)]

p ,const pYGS'cV . (71)

22

Then Eq. (70) becomes [Ref. 18, Eq. (8-3)] for • = 0 and

S = const (isentropic)

, + -v 2 + t- t+ H - a constant • (72)

Therefore irrotational flows are not necessarily isoenergeticin the sense that the total energy H is a constant throughoutthe field. Irrotational flows are isoenergetic only if theyare steady, i.e., 3/3t - 0.

On the other hand for a rotational flow Eqs. (67) and (68)show

that when the motion is steady (i.e., ay/_t - 0) the total

energy H is a constant along any streamline or vortex line(compare with the energy equation Eq. (37)]. The differenca

is that for steady (rotational) isoenergetic flow the motionis not necessarily isentropic, while for steady irrotational

(isoenergetic) flow the motion must be isentropic. An examplefor the isoenergetic but nonisentropic flow is the problem of

steady supersonic flow o•ver a body with curved detached shock.

23

W --'*'*

SECTION 4

TRANSFORMATION OF CONSERVATION EQUATIONS INTERMS OF ORTHOGONAL CURVILINEAR COORDINATES

The vectorial forms of the mass, momentum and energy conserva-tion equations Eqs. (32), (33), and (34) can be expressed in

a general curvilinear coordinate system by substituting theexpressions for the gradient, divergence, and curl operatorsin such a system given, for example, in Reference 20, Appendix C.For practically useful orthogonal curvilinear coordinates x1,

x 2, x 3, with local unit vectors e1, e 2, e3 in the directions

of increase of x1 , x2 , x 3 the specific expressions can be

found in Ref. 21, Appendix 2.

An elementary displacement can be written as

ds = hIdxle1 + h 2 dx 2e 2 + h3 dx3 e 3 (73)

where h1 , h2 and h 3 are the scale factors. The velocity com-ponents are vl, v 2 and v 3 such that

v = vlel + v 2 e2 + v 3 13 (74)

Then Eqs. (32), (33) and (34) can be written as follows

(Ref. 16, Section 1.2.2) :

mass; a + h 1h 3 a~ (1-h h hPv) M.0 *(75)

Bt h ~ h h h1 2 3 sJ ) +

1 (12 3' l ;)2momentum: (Pv1) + h Fj 2h3O i

+ -.+ -1- (Tl 2h T'Xh3 22

4(76)

24

and two morons by cyclic permutation

energy: a 1 a h(

I1 3T

Tijvi - k _ BT f-v + G (77)

where the suconvention has been used and Tij are thecomponents omsor I :

pvv - w+ . (78)

They can be ed as

=pviv + P6ij T ij (79)

with

I(V*v)6ij + 21 Dij (80)

where

(1 = (81)

is the Kronecbol.

In Eq. (80)

h1 2 3 1 ~ h1 h h v )(2

vi v 2 h1 V3 h(83)

25

ne- "p,

with D22 and D3 3 following by cyclic permutation and

ij(ij ax i -2) .ax4

26

4 f.bl

SECTION 5

CONSE EQUATIONS IN CARTESIAN COORDINATES

To the Cazcoordinates

x 1 = X, X2 = y, X3 =Z (85)

there corzthe scale factors

h 1 = h2 = h 3 = 1 (86)

The cons4zequations Eqs. (75), (76) and (77) become

when Eq. (used:

mass: - (Pv ) = 0 , (87)'C. j

Ja

momentum: .) + x- (Pvivj) + " (88)""i

8

energyt + 4 (PE+P)vj -T~V k f~v + GI R ij i 4 - ivi(89)

wheresindqs. (82) to (84)av£

V.v (90)

and

Di + i(91)

we havefg(80),

= a ( 3 (92)Sij rx + ( x +These resme with those usually found in the literature

expressed.sian tensor notations (see, for example,Ref. 9, Ed9)).

27

SECTION 6

CONSERVATION EQUATIONS IN CYLINDRICAL COORDINATES

To the cylindrical coordinates

x1 = r, = e, x 3 = z (93)

there correspond the scale factors (Reference 22, Appendix 4)

hI1 = 1, h 2 = r,, h 3 = 1 .(94)

The conservation equations Eqs. (75), (76) and (77) become

ms + (rpvr) + 5 (pv,) + a (PV) 0, (95)

momentum: • (pVr) + + +(

5 r 22 = fr (6

r (Pvo) + r rr 21) + n (323)

+ _ 21 f 1 (97)

3-E (2vz) + (rf 3 1 ) + (32)

+ a (8+ jz (533) =f (98)

energy: (pE) + 1 'r I E+pr- (TIIVr+T2 iv +T3 1vz)

-k 2

28

+ I- (pE+p)V 0 - (TI 2 Vr+T22V0 +T3 2Vz)

k T

+z 13 +T2vPE+P)vz - (Tl3 r¶2V+¶33vz)

az

=frvr + foVe+ fzvz + G (99)

where 3'ij is given by Eq. (79) with Tij given by Eq. (80)

in which,according to Eqs. (82) to (84),

a (rv) +1 + (100)Tr r T

and

SD11 2 r 7 + 2 0 D33 IF

12 21 " IT -r r (101)

Di (D vz + v 0D23 032 1f 1 - W "a',0

S3 r V z2D31 = 13 2 ½ \"a T r-)/

29

SECTION 7

CONSERVATION EQUATIONS IN SPHERICAL COORDINATES

To the spherical coordinates

xI = r, x2 = 6, x3 =p (102)

there correspond the scale factors (Reference 21, Appendix 2)

h1i = 1, h2 = r , h3 = r sin 6 • (103)

The conservation equations Eqs. (75),, (76) and (77) become

mass: + r 2 sin 7 F (r 2sinOpvr) + a (rsin8opv 0 )

+ (rpv,)] =w 0 (104)

momentum: - (vr) + r 2 (r sin0 ) + (raineis 12 )r a in l

r+ 1 3 ),1 " I 22 rl S33 rf (105)

Oe (Pv) + 1 (r2 sinas,) + •..(rsinOS2 2 )

cose 3 - , (106)

US23 MJ3 - Fo8 r 21 fe

"3E (Pvc) + r-s-.. [ (r siner 3 1 ) + I (rsinS 3 2 )r sino er32

+ ] 1 cosp,(107)

30

energy: PE + 1 r 2 sin (E+p)vr -21e31r 2 sine

k + ir2sin sinel(PE+P)ve- (T. 1 2Vr+22v6+T3 2v•)

k +r~sn1 a r[(pE+p) vv (T Vr+2

r r2sin)

k T f .f v e oV+ f,,pV +G j, (108)

where Tij is given by Eq. (79) with Tij given by Eq. (80),

in whick according to Eqs. (82) to (84),

VV1 (r 2sin0v ) + (rsin~ v + (109)

rVs sine

and

aV r a(3 Vra 3D f- D22 1 + vr t D33 = + sinevr+cCOGO

ive ve + ! aVr\12 D 21 f m r- +r r

D D 1 sn av+ ave (11'

23 322sn YESINO Tr "W

I lav r'., a \

D 31 D13 = 2rsin " + rsainO -r- sine .

31

SECTION 8

COMPACT ONE-DIMENSIONAL FORMULATION

The conservation equations in Sections 5, 6 and 7 for three-

dimensional Cartesian, cylindrical and spherical problems

respectively, can be readily reduced to those for various one-and two-dimensional problems by setting appropriate velocity

components, force components and derivatives to zero. Inparticular, if we consider the one-dimensional problems inwhich

a( ) •()

x r, v = elvr(r), v 2 v 3 0, v = =0 f 2 V f 3 = 0

S~(111)

the conservation equations in Cartesian, cylindrical andspherical coordinates can be compactly expressed in one single

set of equations (with j - 0, 1, 2 representing plane, lineand point symmetry respectively):

maus: + r (rJpvr) -0, (112)- ~at rr

momentum: (ov) + a4 (rJPv 2)

[.V

S.~r ~ i (113)

f r3rrF ar - r Vr ar'(13

energy: (pE) + - [rivr0E+p)I[Tr vr rrv ~

r

+ 1 a (k~r aT G *(114)

32

SECTION 9

CONCLUSIONS

* Following a survey of the basic equations, the mass, momentum

and energy conservation equations and the components of thedeformation and stress tensors for a nonsteady flow of a com-

pressible viscous single-phase fluid have been expressed invector and dyadic notations, transformed in terms of ortho-

gonal curvilinear coordinates and specialized for the three-dimensional cases in Cartesian, cylindrical and spherical

coordinates. These equations caa then be reduced to corres-ponding equations for one- and two-dimensional flows in variouscoordinates. They are in a format readily applicable to

4 practical problems.

If desired the procedure can be extended to general nonortho-

gonal curvilinear coordinate systems.

This report also contains a generalization of the conservationequations in integral form based on an extension of the Reynoldstransport theorem.

33

SECTION 10

LIST OF REFERENCES

1. Truesdell, C., The Kinematics of Vorticity, Indiana Univer-sity Press, Bloo--ngton, 1954.

2. Euler, C., Principes g6n6raux du mouvement des fluides,Hist. Acad. Berlin 1955, pp. 274-315.

3. Serrin, J., "Mathematical Principles of Classical FluidMczhanics," Enc yclopedia of Physics, S. Fliigge (ed.),Vol. VIII/1: Fluid Dynamics I, C. Truesdell (co-editor),Springer-Verlag, Berlin, 1959, pp. 125-263.

4. Reynolds, 0., The Sub-Mechanics of the Universe, Vol. IIIof "Papers on R-echanical and Physical Subjects,"CambridgeUniversity Pross, London, 1903.

5. Boris, J. P., "Flux-Corrected Transport Modules forGeneralized Continuity Equations," NRL Memorandum Report3237, Naval Research Laboratory, Washington, D.C., 1976.(Also see Finite-Difference Techniques for VectorizedFluid Dynamics Calculations edited by D. L. Book, Springer-Verlag, New York, N.Y., 1981, pp. 33-40.)

6. Schlichting, H., Boundary Layer Theor-, 7th Edition,(translated by J. Kestin), McGraw-Hill Book Company, NewYork, U.Y., 1979.

7. Owczarek, J. A., Fundamentals of Gas Dynamics, Interna-tional Textbook Co., Scranton, Pennsylvania-,1964.

8. Shapiro, A. H., The Dynamics and Thermodynamics of Com-•ressibie Fluid Flow, Vols. I and II, Ronald Press, NewYork, N.Y., 1953.

9. Liepmann, H. W. and Roshko, A., Elements of Gas Dynamics,John Wiley & Sons, Inc., New YorkR, N.Y., 5.

10. Pai, S. I., Viscous Flow Theory, (I-Laminar Flow, 1956II-Turbulent Flow, 1957), D. van Nostrand Company, Inc.,Princeton, N.J.

11. von Mises., R., Mathematical Theory of Compressible FluidFlow, (completed by H. Geiringer and G. S. S. Ludford),Aca•emic Press, Inc., New York, N.Y. 1958.

12. Courant, R. and Friedricks, K. 0., Supersonic Flow andShock Waves, Interscience Publishers, Inc., New York, N.Y.,1948.

34

_% %

44.

13. Chapman, A. J. and Walker, W. F., Introductory GasDynamics, Holt, Rinehart and Winston, New York, N.Y., 1971.

14. Roache, P. J., Computational Fluid Dynamics, HermosaPublishers, Albuquerque, New Mexico, 1972.

15. Potter, D., Computational Physics, John Wiley & Sons,New York, N.Y., 1973.

16. Peyret, R. and Taylor, T. D., Computational Methods inFluid Flow, Springer-Verlag, New York, N.Y., 1983.

17. Milne-Thomson, L. M., Theoretical Hydrodynamics, 4th ad.,The Macmillan Company, New York, N.Y., 1960

18. Tsien, H. S., "The Equations of Gas Dynamics," Fundamentalof Gas Dynamics, H. W. Emmons (ed.), Vol. III: High SpeedAodynamiCs ad Jet Propulsion, Princeton UniversityPress, Princeton, N.J., 1958.

19. Slattery, J. C., Momentum, Energy and Mass Transfer in* Continua, McGraw-Hill Book Company, New Yori,, N.I, 12'

20. Eringen, A. C., Mechanics of Continua, John Wiley & Sons,Inc., New York, N.Y-., 96..

21. Batchelor, G. K., An Introduction to Fluid DnamicsCambridge University Press, London, 1967.

22. Lebedev, N. N., Skalskaya, I. P., and Uflyaad, Y. S.,Problems of Mathematical Physics, R. A. Silverman (trans-lator and editor), Prentice-Hall, Inc., Englewood Cliffs,N.J., 1965.

35

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36

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