Conservation equations of viscous heat conducting fluid in curvilinear orthogonal coordinate

Black, L. H.

English

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NATIONAL. AERONAUTICS AND SPACE ADMINISTRATION 
Technical Report 32-1332 
Conservation Equations of a Viscous, 
Heat-Conducting Fluid in 
Curvilinear Orthogonal 
Coordinates 
L./-/. Back 
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JET PROPULSION LABORATORY 
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 
September 15, 1968
Oct I 
iSP 41 
https://ntrs.nasa.gov/search.jsp?R=19680025794 2020-03-12T06:51:37+00:00Z
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
Technical Report 32-1332 
Conservation Equations of a Viscous, 
Heat-Conducting fluid in 
Curvilinear Orthogonal 
Coordinates 
L. H. Back 
Approved by: 
jel^l 
artz,Mane 
Research and Advanced Concepts Section 
JET PROPULSION LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA
September 15, 1968
TECHNICAL REPORT. 32-1332 
Copyright© 1968
Jet Propulsion Laboratory
California Institute of Technology 
Prepared Under Contract No. NAS 7-100 
National Aeronautics & Space Administration
Contents 
I. Introduction	 ......................... 1 
II. Conservation Equations	 ....................1 
Nomenclature	 .........................6 
References	 ..........................7 
Table 
1. Coordinate systems and scale factors ................4 
JPL TECHNICAL REPORT 32-1332 	 iii
Abstract 
A complete, written set of the conservation equations of a viscous, heat-
conducting fluid is given in curvilinear orthogonal coordinates. Scale factors for 
a number of coordinate systems are tabulated for convenience in expressing the 
equations in various coordinates. 
iv	 JPL TECHNICAL REPORT 32-1332
Conservation Equations of a Viscous, Heat-Conducting 
Fluid in Curvilinear Orthogonal Coordinates 
I. Introduction 
Although formulation of the conservation equations of 
a viscous, heat-conducting fluid in curvilinear orthogonal 
coordinates is well known through vector and tensor 
analysis (Refs. 1 and 2), a complete, written-out set of 
equations, including the energy equation, is not readily 
available in any given source. The momentum equation 
was given by Goldstein (Ref. 3) in curvilinear orthogonal 
coordinates for an incompressible, constant-property fluid, 
and by Tsien (Ref. 4) for a compressible, variable-property 
fluid. Only the commonly used special cases of the set of 
equations in rectangular, cylindrical, and spherical co-
ordinates appear in the literature (e.g., Ref. 5). The pur-
pose of this paper is to briefly present the complete 
set of equations in stationary, curvilinear orthogonal co-
ordinates. For convenience in expressing the equations in 
various coordinates, scale factors for eleven coordinate 
systems are tabulated. 
II. Conservation Equations 
Three forms of the energy equation are considered, one 
form of which may be best suited for a particular appli-
cation. These relations involve the total enthalpy H,
p+p(VV)'Ht =	 - Vq+ V(TV) 
at	 at
+ FV + W 
the internal energy E, 
aE 
p+p(VV)E+pVV —Vq+r:(VV)±W 
and the enthalpy H, 
-; +p(V . V)H—[ +(V.V)p] 
—
V q+ i-: (VV) + W 
To tomplete the set of conservation equations, the con-
tinuity and momentum equations are, respectively, 
ap + V-(PV) =0 
at 
V I	 1	 Fl 
at	 P	 p	 p 
The quantities that appear in these equations are identi-
fied in the nomenclature. 
JPL TECHNICAL REPORT 32-1332
By use of the same notation as used by Goldstein 
(Ref. 3), the orthogonal coordinates are taken as a, /3, and 
y such that the elements of length at a, /3, and y in the 
directions of increasing a, 6, and y are h1da, h.df3, and 
h3d-y, respectively. The differential are length ds is, then, 
- 
h (da)2 + h (d/3) 2
 + hl (dy)2 
If u, v, and w are components of the velocity vector V in 
direction of increasing a, /3, and y, the continuity equa-
tion is 
ap+ 1 
at	 h1h2h3 
	
>< [- (h2h3pu) +
	 (hh3pv) + - (h1h2pw) ] =
ay 
The momentum equation written in the a, /3, and y direc-
tions is 
au	 u 
all
v a	 w au	 uv all,	 uw 
	
v 2 h2	 w2 h3	 1 1 ap Fa	 1 
-	 -	 - -	 + - + - (V r)a hill ., va 	 va	 p n1 va	 p	 p 
av	 av V av w av	 vu 
U2 all,
	 w2 h3 - -	
+	 + ( V - hill,- -- - •i	
-	
h2 ap	 p	 p 
aw	 a	 , a	 a	 wu h,	 ak 
	
U2 h1	 v2 h2 - -
	 + + (V 
-	
_;- - •i;;:i -;:;;- -	 , h3 ay	 p	 p 
The components of the divergence of the symmetric 
viscous stress tensor r in the x, /3, and y direction (Ref. 6)1 
'The h,, h, and h, used by Love are the reciprocals of those used 
herein.
are: 
(Vr)a 
	
h1hh3 [ (hh3 Tra) +	 (hih3 ran) +	 (h. It, ra)] a7 
1h1	 I 
+	 + Ty 
•h,h3 a 
1 all,	 1 all, 
— T-- —
a-
( 17- r) = 
	
1 [(h2h3 rae) +	 (h 1 h 1	 h +	 (h ru)] h1 h 2h3 ac,
1 h.	 1	 h. 
+	 + Tfi h1h. aa
	 h2h3 a7 
1 all,	 1 all, 
— Taa7	 Th0h/3 
(Vr)y= 
	
h1h.,h.1 [- (h2 h3 r Ya) +	 (h 1 h3	 ) +	 (h1h2 
1	 h3	 1 h. 
+ WT, —a + 
1 all,	 1 a ll, 
— Taa	 — Tf3	 , 
1	 a7 	 (7), 
The components of the viscous stress tensor for a Stokes' 
fluid are related to the components of the rate of strain 
tensor by
Taa AVV+eaa 
Too = xV V + 
Tyy AVV+e 
	
Ty	 Ta 
Tay = 
Tj'3'f = Typ = 
where the divergence of the velocity vector is 
V V 
= h1h2h3 [- (h2h3u) +	 (h1 h3v) + -i-- (hih2w)] 
2	 JPL TECHNICAL REPORT 32-1332
and the components of the rate of strain tensor are 
(Ref. 3):
The rate at which work is done by body forces is, simply, 
1	 1 au	 V ah1 	 w 
-- e, = - - + hja 
1	 1 av	 W ah	 ah2 
1	 law	 u ah,	 ah3 
h., a	 h1 
eai=(r)+——) 
h 1 a	 h3a 
h3 a (w) 
+ h
2 	 fv )
e(7 =  
The second viscosity coefficient A is related to the shear 
viscosity A (first viscosity coefficient) by A = — 2/3 [t if the 
bulk viscosity coefficient defined by K = A + 2/3 is zero. 
Otherwise, A is given by
2 
A = K - 
In the various forms of the energy equations, the oper-
ator (V . V) applied to a scalar f, such as H, E, p, or H, 
gives the convection of that quantity by the flow, 
1 af	 1 af	 laf(V V)f =u j —— +v+Wr a) 
The divergence of the heat flux vector q is 
V q 
= h1h2h. [- (h 2 h3qa) +	 (h 1 h3q) + -- (hih2q)] 
In particular, if the heat flux vector is given by Fourier's 
heat-conduction law, q = — k V T, then the components 
are
1 aT	 1 aT 
qa= — k - --,	 q=—k-j---, 
q-1 = —
k 1 T 
-;
F V = Fc,u + Fv + F7w 
The rate at which work is done by the viscous stresses is 
given by
í a V (r V)	 1 h1h2h3 - [h 2h3 (raaU + TV + T7aW)] 
+	 [h1h3 (raU + rflV ± T-)q3W)] 
ap 
+	 [hh2 (rayu + r-yV + 
ay	 f 
Lastly, the rate of dissipation of energy takes the form 
1 a v
--+ 
+ T$$ (- 
l	 ay u	 h.,	 w 
±	 + hh 
/ 1 aw 
+r + +i 
ah,,	 v 
/l au lay	 v ah 2 	 u 
+Ta + 
'
+	 (h, 
1 au 1 aiv	 w 
--- --
u
--
y h,	 a	 h 1 h.
---
h aa	 h1 ay 
/ 1 av 1 aw	 w vh3. 
This rate of dissipation of energy term usually appears in 
the literature as D. 
Table 1 presents descriptive information on a number 
of orthogonal coordinate systems for which the conserva-
tion equations can be readily written by use of the fore-
going relations. The last two entries in Table 1, in which 
the coordinates are taken along and normal to the surface, 
are useful in analyzing internal and external boundary-
layer flows. For many flow problems in these coordinates, 
the dominant viscous stress is the shear stress that lies in 
the plane of f3 = const (TIP for a two-dimensional flow and 
rae, T.0 for a three-dimensional flow), and the important 
heat-flux component is normal to the surface, q. 
JPL TECHNICAL REPORT 32-1332 	 3 
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Nomenclature 
e1 , = components of rate of strain tensor T = temperature 
E = internal energy per unit mass u = velocity component in a direction 
f = scalar v = velocity component in p direction 
F = body force per unit volume
V = velocity vector 
h1 h2h3 = scale factors
w = velocity component in	 direction 
H = static enthalpy per unit mass
W	 heat generation per unit volume 
total enthalpy, H = H + V2/2 
k = thermal conductivity a,	 y = orthogonal coordinates 
p	 static pressure K = bulk viscosity 
q = heat-flux vector A = second viscosity coefficient 
qp = heat-flux component normal to pt= shear viscosity
 
the surface p = density 
t = time = viscous stress tensor
6	 JPL TECHNICAL REPORT 32-1332 
References 
1. Lagerstrom, P. A., "Laminar Flow Theory," Theory of Laminar Flows, Vol. IV, 
High-Speed Aerodynamics and Jet Propulsion, ed. by F. K. Moore. Princeton 
University Press, 1964. 
2. Morse, P. M., and Feshbach, H., Methods of Theoretical Physics, Part I. 
McGraw-Hill Book Co., Inc., New York, 1953. 
3. Goldstein, S., Modern Developments in Fluid Dynamics, Vol. I, Oxford Uni-
versity Press, 1938. 
4. Tsien, H. S., "The Equations of Gas Dynamics," in Fundamentals of Gas Dy-
namics, Vol. III, High-Speed Aerodynamics and Jet Propulsion, ed. by H. W. 
Emmons. Princeton University Press, 1958. 
5. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena. J. Wiley 
& Sons, Inc., New York, 1960. 
6. Love, A. E. H., Treatise on the Mathematical Theory of Elasticity, p. 90. Dover 
Publications, New York, 1st American Printing, 1944. 
JPL TECHNICAL REPORT 32-1332	 7


Conservation Equations of a Viscous, Heat-Conducting Fluid in Curvilinear Orthogonal Coordinates

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