The explicit nonlinearity permeating Navier–Stokes continuum conservation law statements has challenged fluid mechanics analytical characterization since topic dawning. In testimony to and in recognition of the contributions of Professor Frank M. White to subject knowledge promulgation, this review traces evolution of fluid mechanics mathematical insight successes from earliest times through the half century of computational fluid dynamics (CFD) theorization evolution to present day identification of a totally analytical “mathematics of physics of fluids” spatially filtered Navier–Stokes CFD algorithm/code amenable theorization.

1.
Amsden
,
A.
, and
Harlow
,
F.
, “
The SMAC method: A numerical technique for calculating incompressible fluid flow
,”
LASL Technical Report No. LA-4370
(University of California,
1970
).
2.
Anderson
,
D. A.
,
Tannehill
,
J. C.
, and
Pletcher
,
R. H.
,
Computational Fluid Dynamics and Heat Transfer
(
Hemisphere Publishing
,
New York
,
1984
).
3.
Babuska
,
I.
, and
Rheinboldt
,
W. C.
, “
A-posteriori error estimates for the finite element method
,”
Int. J. Num Methods Eng.
12
,
1597
(
1976
).
4.
Baker
,
A. J.
,
Finite Element Computational Fluid Mechanics
(
Hemisphere Publishing
,
New York
,
1983
).
5.
Baker
,
A. J.
, and
Kim
,
J. W.
, “
A Taylor weak statement algorithm for hyperbolic conservation laws
,”
Int. J. Numer. Methods Fluids
7
,
489
(
1987
).
6.
Baker
,
A. J.
,
Optimal Modified Continuous Galerkin CFD
(
Wiley
,
London
,
2014
).
7.
Baker
,
A. J.
,
CFD Error Elimination with Optimal Finite Element Constructions
(
Elsevier
,
London
,
2021
).
8.
Batchelor
,
G. K.
,
An Introduction to Fluid Dynamics
(
Cambridge University Press
,
Cambridge
,
1967
).
9.
Beam
,
R.
, and
Warming
,
R.
, “
An implicit finite-difference algorithm for hyperbolic systems in conservation law form
,”
J. Comput. Phys.
22
,
87
(
1976
).
10.
Blasius
,
H.
, “
Grenzschichten in Flussigkeiten mit kleiner Reibund
,”
Z. Math. Phys
56
,
1
(
1908
).
11.
Button
,
K.
, “
Decoding the boundary layer at hypersonic speeds
,”
Aerospace America
57
,
9
(
2019
).
12.
Courant
,
R.
,
Friedrichs
,
K.
, and
Loewy
,
H.
, “
Uber die Partiellen Differenzengleichungen der Mathematischen Physik
,”
Math. Ann.
100
,
32
(
1928
).
13.
Crank
,
J.
, and
Nicolson
,
P.
, “
A practical method for numerical evaluation of solutions of partial differential equations of heat-conduction type
,”
Math. Proc. Cambridge Philos. Soc.
43
,
50
(
1947
).
14.
Dorrance
,
W. H.
,
Viscous Hypersonic Flow
(
McGraw Hill
,
New York
,
1962
).
15.
Fabrick
,
A.
,
Sklarew
,
R.
, and
Wilson
,
J.
, “
Point source model evaluation and development study
,”
Technical Report No. A5-058-87
(
Science Applications Corporation
,
1977
).
16.
Finlayson
,
B. A.
,
The Method of Weighted Residuals and Variational Principles
(
Academic Press
,
New York
,
1972
).
17.
Galdi
,
G. P.
, and
Layton
,
W. J.
, “
Approximation of the larger eddies in fluid motion II: A model for space filtered flow
,”
Math. Models Methods Appl. Sci.
10
,
343
(
2000
).
18.
Galerkin
,
B. G.
, “
Series occurring in some problems of elastic stability of rods and plates
,”
Eng. Bull.
19
,
14
(
1915
).
19.
Georgiadis
,
N. J.
,
Yoder
,
D. A.
,
Towne
,
C. S.
,
Engblom
,
W. A.
,
Bhagwandin
,
V. A.
,
Power
,
G. D.
,
Lankford
,
D. W.
, and
Nelson
,
C. C.
, “
Wind-US code physical modeling improvements to complement hypersonic testing and evaluation
,”
NASA Technical Report No. TM-2009-215615
(
2009
).
20.
Godunov
,
S. K.
, “
Finite-difference method for numerical computation of discontinuous solution of the equations of fluid dynamics
,”
Mat. Sb.
47
,
271
(
1959
).
21.
Gonzalez
,
D.
, and
Medina
,
A.
, “
Experimental-numerical campaigns give insight into complex fluid phenomena
,”
Aerospace America
57
,
11
(
2019
).
22.
Gosman
,
A.
,
Pun
,
W.
,
Runchal
,
A.
,
Spalding
,
D.
, and
Wolfshtein
,
M.
,
Heat and Mass Transfer in Recirculating Flows
(
Academic Press
,
London
,
1969
).
23.
Hafez
,
M.
,
South
,
J.
, and
Murman
,
E.
, “
Artificial compressibility methods for numerical solution of transonic full potential equation
,”
AIAA J.
17
,
838
(
1979
).
24.
Harlow
,
F. H.
, and
Welch
,
J. E.
, “
Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface
,”
Phys. Fluids
8
(
12
),
2182
(
1965
).
25.
Harlow
,
F.
, “
Fluid dynamics: A LASL monograph
,”
LASL Technical Report No. LA-4700
(
1971
).
26.
Hayes
,
W. D.
, and
Probstein
,
R. F.
,
Hypersonic Flow Theory, Volume 1, Inviscid Flows
, 2nd ed. (
Academic Press
,
New York
,
1966
).
27.
Jameson
,
A.
, “
Iterative solution of transonic flows over airfoils and wings including flows at Mach 1
,”
Commun. Pure Appl. Math.
27
,
283
(
1974
).
28.
John
,
V.
, “
Large eddy simulation of turbulent incompressible flows, analytical and numerical results for a class of LES models
,”
Lecture Notes in Computational Science and Engineering
(
Springer
,
Berlin
,
2004
), Vol.
34
.
29.
John
,
V.
, “
An assessment of two models for the subgrid scale tensor in the rational LES model
,”
J. Comput. Appl. Math.
173
,
57
(
2005
).
30.
Kolesnikov
,
A.
, “
Efficient implementation of high order methods in computational fluid dynamics
,” Ph.D. dissertation (
University of Tennessee
,
2000
).
31.
Kolesnikov
,
A.
, and
Baker
,
A. J.
, “
An efficient high order Taylor weak statement for the Navier-Stokes equations
,”
J. Comput. Phys.
173
,
549
(
2001
).
32.
Lax
,
P.
, “
Weak solutions of nonlinear hyperbolic equations and their numerical computation
,”
Commun. Pure Appl. Math.
7
,
159
(
1954
).
33.
Lax
,
P.
, and
Wendroff
,
B.
, “
Systems of conservation laws
,”
Commun. Pure Appl. Math.
13
,
217
(
1960
).
34.
Legensky
,
S.
, “
Why we're not there yet with CFD
,”
Aerospace America
22
,
101089
(
2021
).
35.
Le Quere
,
P.
, and
Benhia
,
M.
, “
From onset of unsteadiness to chaos in a differentially heated square cavity
,”
J. Fluid Mech.
359
,
81
(
1998
).
36.
MacCormack
,
R.
, “
The effect of viscosity in hypervelocity impact cratering
,” AIAA Paper No. 69-354,
1969
.
37.
Maskew
,
B.
, “
A computer program for calculating the nonlinear aerodynamic characteristics for arbitrary configurations
,” NASA Contract Report No. CR-166476 (
1982
).
38.
Masters
,
J.
, “
A year of enhancements, plus a groundbreaking vision document
,”
Aerospace America
52
(
11
),
26
(
2014
).
39.
Murman
,
E. M.
, and
Cole
,
J. D.
, “
Calculation of plane steady transonic flow
,”
AIAA J.
9
,
114
(
1971
).
40.
NAP
,
Assessing the Reliability of Complex Models: Mathematical and Statistical Foundations of Verification, Validation, and Uncertainty Quantification
(
National Academies Press
,
Washington DC
,
2012
).
41.
Nikuradze
,
J.
, “
Gesetzmassigkeiten der turbulenten Stromung in glatten Rohren
,”
VDI-Forschungsh.
1932
,
356
.
42.
Pope
,
S. B.
,
Turbulent Flows
(
Cambridge University Press
,
New York
,
2000
).
43.
Prandtl
,
L.
,
Essentials of Fluid Dynamics
(
Hafner
,
New York
,
1952
).
44.
Richtmyer
,
R. D.
, and
Morton
,
K. W.
,
Difference Methods of Initial-Value Problems
, 2nd ed. (
Interscience Publishers
,
New York
,
1967
).
45.
Roache
,
P. J.
,
Computational Fluid Mechanics
(
Hermosa
,
Albuquerque
,
1972
).
46.
Roe
,
P. L.
, “
Approximate Riemann solvers, parameter vectors and difference schemes
,”
J. Comput. Phys.
43
,
357
(
1981
).
47.
Saad
,
Y.
, and
Schultz
,
M. H.
, “
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
,”
SIAM J. Sci. Stat. Comput.
7
,
856
(
1986
).
48.
Sahu
,
S.
, and
Baker
,
A. J.
, “
A modified conservation principles theory leading to an optimal Galerkin CFD algorithm
,”
J. Numer. Methods Fluids
55
(
8
),
737
(
2007
).
49.
Schlichting
,
H.
,
Boundary Layer Theory
(
McGraw-Hill
,
New York
,
1979
).
50.
See
Jameson
,
A.
, http://dept.ku.edu/~cfdku/JRV/Jameson.pdf for “
Reflections on Four Decades of CFD: A Personal Perspective
” (
2012
).
51.
Sekachev
,
M. A.
, “
Essentially analytical theory closure for thermal-incompressible space filtered Navier-Stokes PDE system
,” Ph.D. dissertation (
University of Tennessee
,
2013
).
52.
Sekachev
,
M. A.
,
Baker
,
A. J.
, and
Wong
,
K. L.
, “
Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number, I. Theory, II. Resolutions, validations, III. FaNS theory validation
,”
J. Numer. Heat Transfer, Part B
70
(
4
),
267
(
2016
).
53.
Shames
,
I. H.
,
Mechanics of Fluids
(
McGraw-Hill
,
New York
,
1962
).
54.
Shapiro
,
A. H.
,
The Dynamics and Thermodynamics of Compressible Fluid Flow
(
Ronald Press
,
New York
,
1954
), Vol. 2.
55.
Slotnick
,
J.
,
Khodadoust
,
A.
,
Alonso
,
J.
,
Darmofal
,
D.
,
Gropp
,
W.
,
Lurie
,
E.
, and
Mavriplis
,
D.
, “
CFD vision 2030 study: A path to revolutionary computational aerosciences
,” “
NASA Technical Report No.
CR-2014-218178 (
2014
).
56.
Stumpe
,
J.
, “
Why CFD and wind tunnels need each other
,”
Aerospace America
56
(
6
),
30
(
2018
).
57.
Tannehill
,
J. C.
,
Anderson
,
D. A.
, and
Pletcher
,
R. H.
,
Computational Fluid Dynamics and Heat Transfer
, 2nd ed. (
Taylor & Francis
,
Washington DC
,
1997
).
58.
Thompson
,
J. F.
,
Warzi
,
Z. U. A.
, and
Mastin
,
C. W.
,
Numerical Grid Generation, Formulations & Applications
(
Elsevier
,
New York
,
1985
).
59.
Van Dyke
,
M.
,
Perturbation Methods in Fluid Mechanics
(
Academic Press
,
New York
,
1964
).
60.
van Leer
,
B.
, “
Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's method
,”
J. Comput. Phys.
32
,
101
(
1979
).
61.
White
,
F. M.
,
Viscous Fluid Flow
(
McGraw-Hill
,
New York
,
1974
).
62.
White
,
F. M.
,
Fluid Mechanics
(
McGraw-Hill
,
New York
,
1978
).
63.
Wilcox
,
D. C.
,
Turbulence Modeling for CFD
(
DCW Industries
,
La Canada, CA
,
2006
).
64.
Williams
,
P. T.
, “
A 3D time-accurate incompressible Navier-Stokes finite element CFD algorithm
,” Ph.D. dissertation (
University of Tennessee
,
1993
).
65.
Williams
,
P. T.
, and
Baker
,
A. J.
, “
Incompressible computational fluid dynamics and the continuity constraint method for the 3D Navier-Stokes equations
,”
J. Numer. Heat Transfer, Part B
29
,
137
(
1996
).
66.
Xin
,
S.
, and
Le Quere
,
P.
, “
Linear stability analysis of natural convection flowfields in a differentially heated square cavity with conducting walls
,”
Phys. Fluids
13
,
2529
(
2001
).
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