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CHAPTER V: VERIFICATION PROBLEMS
KNOWN ANALYTIC SOLUTIONS

Shortcuts:

Problem V.01: FULLY DEVELOPED LAMINAR FLOW IN A CHANNEL

Problem V.02: FULLY DEVELOPED LAMINAR FLOW IN A PIPE - Re_# = 100

Problem V.03: DEVELOPING LAMINAR FLOW IN A CHANNEL - RE = 150

Problem V.04: DEVELOPING LAMINAR FLOW IN A PIPE - RE = 150

Problem V.05: Boundary LAYER ON A FLAT PLATE WITH ZERO dP/dX

Problem V.06 - SOLID BODY ROTATION >>> C37

Problem V.07 - CORNER FLOW >>> C38

Problem V.08 - DIFFERENTIAL ROTATION >>> C39

Problem V.09 - SWIRLING FLOW >>> C52

Problem V.10 - SWIRLING FLOW >>> C53

Problem V.11 - RADIAL FLOW >>> C54

Problem V.12 - AXIAL FLOW >>> C55

Problem V.13 - AXIAL POISEUELLE FLOW >>> C56

Problem V.14 - LAMINAR FULLY DEVELOPED FLOW IN A PIPE >>> C57

Problem V.15: CONDIF TEST CASE FOR STEP FLOW - theta = 45 deg

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************************************************************************
TITLE Problem V.02: FULLY DEVELOPED LAMINAR FLOW IN A PIPE - Re_# = 100
************************************************************************
/
GRID NODEs 42 by 22
/
COORDINATE X RANGE = 20.0, RATIO = 1.02
COORDINATE R RANGE = 1.00, RATIO = 1.00
/
WALL at undefined outer boundaries
INLEt at X- boundary
OUTLet at X+ boundary
SYMMetry at Y- boundary
/
SET POWER LAW: U = -2.0 *(Y +0.)^ 2.0 +2.0
LAMINAR FLOW
VISCOSITY 0.01
/
DIAGNOSTIC NODE U V P RP (21,02) print every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (4,2)
OUTPut U, V, P for SELEcted window
SAVE OFF U,V,P ON 'V02.SAV'
/
MATRIX SWEEPS in Y direction only
CONVERGENCE by GLOBAL mode to epsilon = 1.E-9
/
SOLVE for 1000 STEPS in STEADY MODE
/
END
/

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************************************************************************
TITLE Problem V.03: DEVELOPING LAMINAR FLOW IN A CHANNEL - RE = 150
*** Schilichting, 1968 (6th Edition), Page 177
*** Gosman et al., 1969, Page 150;
*** Wang & Longwell, 1964. A.I.Ch.E., 10, 3, p. 323-329.
************************************************************************
/
GRID NODEs 22 by 12
/
COORDINATE X RANGE = 7.0, RATIO = 1.02
COORDINATE Y RANGE = 0.5
/
WALL at undefined outer boundaries
INLEt at X- boundary
OUTLet at X+ boundary
SYMMetry at Y- boundary
/
SET U = 1.000 EVERYWHERE
/
LAMINAR FLOW
VISCOSITY 6.666667E-3
/
DIAGNOSTIC NODE U V P RP (21,02) print every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (2,1)
OUTPut U, V, P for SELEcted window
/
MATRIX SWEEPS in Y direction only
CONVERGENCE by GLOBAL mode to epsilon = 1.E-6
/
SOLVE for 700 STEPS in STEADY MODE
/
SAVE OFF U,V,P ON 'V03.SAV'
/
END
/

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************************************************************************
TITLE Problem V.04: DEVELOPING LAMINAR FLOW IN A PIPE - RE = 150
************************************************************************
/
GRID NODEs 22 by 12
/
COORDINATE X RANGE = 7.0, RATIO = 1.02
COORDINATE R RANGE = 0.5, RATIO = 1.00
/
WALL at undefined outer boundaries
SYMMetry at Y- boundary
INLEt at X- boundary
OUTLet at X+ boundary
/
SET U = 1.000 EVERYWHERE
/
LAMINAR FLOW
VISCOSITY 6.666667E-3
/
DIAGNOSTIC NODE U V P RP (21,02) print every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (2,2)
OUTPut U, P for SELEcted window
SAVE OFF U,V,P ON 'V04.SAV'
/
MATRIX SWEEPS in Y direction only
CONVERGENCE by GLOBAL mode to epsilon = 1.E-6
/
SOLVE for 250 STEPS in STEADY MODE
/
END
/

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************************************************************************
TITLE Problem V.05: Boundary LAYER ON A FLAT PLATE WITH ZERO dP/dX
************************************************************************
/
GRID NODEs 18 by 18
/
COORDINATE X RANGE 20, RATIO 1.2
/
WALL at undefined outer boundaries
INLEt at X- boundary
OUTLet at X+ boundary
OPEN boundary at Y+
/
SET U = 1.0 everywhere
SET P = 0. everywhere
BOUNdary P: Y+: value = 0
/BOUNdary Y+: GRAD V = 0 $ dV/dy = 0. at free boundary
/
LAMINAR FLOW
VISCOSITY VALUE =1.E-3
/
DIAGNOSTIC NODE U V P RP (10,2) print every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELect window from (1,1) to (18,18) skip (3, 2)
OUTPut in NARRow mode for SELEcted window
SAVE OFF U,V,P, ON 'V05.SAV'
/
MATRIX SWEEPS in Y direction only
CONVERGENCE by GLOBAL mode to epsilon = 1.E-6
/
SOLVE for 1000 steps in STEADY mode
/
END
/

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************************************************************************
TITLE Problem V.06 - SOLID BODY ROTATION >>> C37
************************************************************************
/
GRID NODEs 13 by 13
COORdinate corners (0.,0.) (1.,0.) (0.,1.) (1.,1.)
/
SET U LINE Y 0.0 4.0
SET V LINE X 0.0 -4.0
/
DENSity 10.0
VISC 100.0
/
RELA U=0.3, V=0.3, P=0.1
DIFF include SECOnd order SKEW terms for U V
/
DIAGnostic output U V P RP at (5,5) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (2,2)
OUTPut U V FF for SELEcted window
/
CONV GLOB 1.E-6
SOLVE FOR 1000 STEPS IN STEADY MODE
/
SET FF POLYnomial in R: coefficients: 0 0 80 0 0
SET FF LINEar function: 0 -1 * P ADD to existing
SAVE OFF U V P FF on file 'V06.SAV'
/
END
/

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************************************************************************
TITLE Problem V.07 - CORNER FLOW >>> C38
************************************************************************
/
GRID NODEs 13 by 13
COORdinate corners (0.,0.) (1.,0.) (0.,1.) (1.,1.)
/----------------------------------------------------------------------/
/ Initial and Boundary Conditions
/
SET U LINE X 0.0 4.0
SET V LINE Y 0.0 -4.0
///////////////////////////////////// Fluid Properties and Constants
DENSity 10.0
VISC 10.0
/
RELA U=0.3, V=0.3, P=0.1
DIFF SECO SKEW U V W
///////////////////////////////////// OUTPUT CONTROL
DIAGnostic output U V P RP at (5,5) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (2,2)
OUTPut U V FF for SELEcted window
SAVE OFF U V P FF on file 'V07.SAV'
/
///////////////////////////////////// OPERATIONAL CONTROL
CONV GLOB 1.E-10
SOLVE for max of 1000 steps in STEAdy mode
/
///////////////////////////////////// POST-SOLUTION OPERATIONS
SET FF POLYnomial in R: coefficients: 0 0 -80 0 0
SET FF LINEar function: 0 -1 * P ADD to existing
/
END
/
/

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************************************************************************
TITLE Problem V.08 - DIFFERENTIAL ROTATION >>> C39
************************************************************************
/
GRID NODEs 22 by 22
COORdinate corners are: LowerLeft (0.707106781, 0.707106781)
LowerRight (1.673032607, 0.448287736)
UpperLeft (0.448287736, 1.673032607)
UpperRight (1.414213562, 1.414213652)
/
/ Initial and Boundary Conditions
SET U POWEr law 1 * ( R + 0. ) ^ (-2) - 1.
SET U LINEar 0. -0.6666667 * Y MULTiply existing value
SET V POWEr law 1 * ( R + 0. ) ^ (-2) - 1.
SET V LINEar 0. +0.6666667 * X MULTiply existing value
/
/ Fluid Properties and Constants
DENSity 1.0
VISC 0.001
/
/ SOLUTION OPTIONS
RELA U=0.3, V=0.3, P=0.1
DIFF SECO SKEW U V W
/
DIAGnostic output U, V, P at (11,11) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (2,2)
OUTPut FF, FH, FO for SELEcted window in NARRow mode
SAVE OFF U V P, FF, FH, FO on file 'V08.SAV'
/
/ OPERATIONAL CONTROL
CONV GLOB 1.E-6
SOLVE only FOR 200 STEPS IN STEADY MODE
/
//// Analytic Solution: FF = U, FH = V, FO = P
SET FF POWEr law 1 * ( R + 0. ) ^ (-2) - 1.
SET FF LINEar 0. -0.6666667 * Y MULTiply existing value
SET FF LINEar function: 0. -1. * U ADD to existing
SET FH POWEr law 1 * ( R + 0. ) ^ (-2) - 1.
SET FH LINEar 0. -0.6666667 * Y MULTiply existing value
SET FH LINEar function: 0. -1. * V ADD to existing
SET FO 0.888888889 * POWER ( R + 0. )^(2)
SET FO -0.888888889 * POWER ( R + 0. )^(-2) ADD to existing value
SET FO -3.555555556 * LOG ( 1. * R ) ADD to existing value
SET FO LINEar function: 0. -1. * P ADD to existing
/
END
/
/

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************************************************************************
TITLE Problem V.09 - SWIRLING FLOW >>> C52
***** W (ANGULAR VEL) = OMEGA * R
***** P = PREF + 0.5 * RHO * OMEGA**2 * (R**2 -RREF**2)
************************************************************************
/
GRID NODEs 42 by 42
COOR CORN (1,1) (2,1) (1,2) (2,2) CYLI
WALL at undefined outer boundaries
/----------------------------------------------------------------------/
/ Initial and Boundary Conditions
BOUN W VALU LINE Y X- 0.0 10.0
BOUN W VALU LINE Y X+ 0.0 10.0
BOUN W VALU LINE Y Y- 0.0 10.0
BOUN W VALU LINE Y Y+ 0.0 10.0
/----------------------------------------------------------------------/
/ Fluid Properties and Constants
DENSity 2.0
VISC 1.0
SPEC 1.0
PRAN EFFE 0.7
/----------------------------------------------------------------------/
/ SOLUTION OPTIONS
DIFF SECO SKEW U V W P
/
DIAGnostic output U V W P RW at (6,6) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (4,2)
OUTPut for SELEcted window
SAVE OFF U V W P on file 'V09.SAV'
/----------------------------------------------------------------------/
/ OPERATIONAL CONTROL
CONV GLOB 1.E-7 1
SOLVE U V W P STEADY 2000
/
END
/

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************************************************************************
TITLE Problem V.10 - SWIRLING FLOW >>> C53
***** W (ANGULAR VEL) = OMEGA * R
***** P = PREF + 0.5 * RHO * OMEGA**2 * (R**2 -RREF**2)
************************************************************************
/
GRID NODEs 3 by 22 by 22
COOR CORN (0.,1.,0.) (1.,1.,0.) (0.,2.,0.) (1.,2.,0.)
(0.,1.,.7) (1.,1.,.7) (0.,2.,.7) (1.,2.,.7) CYLI
/----------------------------------------------------------------------/
/ Initial and Boundary Conditions
SYMM X-
SYMM X+
BOUN W VALU LINE R Y- 0.0 10.0
BOUN W VALU LINE R Y+ 0.0 10.0
BOUN W VALU LINE R Z- 0.0 10.0
BOUN W VALU LINE R Z+ 0.0 10.0
/----------------------------------------------------------------------/
/ Fluid Properties and Constants
DENSity 1.0
VISC .01
SPEC 1.0
PRAN EFFE 0.7
/----------------------------------------------------------------------/
/ SOLUTION OPTIONS
DIFF SECO SKEW U V W P
RELAX U=0.4
/
DIAGnostic output U V W P RP at (2,6,6) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (2,1,1) to (2,999,999) interval (1,2,2)
OUTPut V,W,P for SELEcted window by YZ planes
SAVE OFF U V W P on file 'V10.SAV'
/----------------------------------------------------------------------/
/ OPERATIONAL CONTROL
CONV GLOB 1.E-10 1
SOLVE U V W P STEADY 1000
/
END
/

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************************************************************************
TITLE Problem V.11 - RADIAL FLOW >>> C54
***** V (RADIAL VEL) = A / R
***** P = PREF + 0.5 * RHO * A**2 * (1/RREF**2 - 1/R**2)
************************************************************************
/
GRID NODEs 42 by 42
COOR CORN (1,1) (2,1) (1,2) (2,2) CYLI
/----------------------------------------------------------------------/
/ Initial and Boundary Conditions
BOUN V VALU POWE Y X- 10.0 0.0 -1.0 0.0
BOUN V VALU POWE Y X+ 10.0 0.0 -1.0 0.0
BOUN V VALU POWE Y Y- 10.0 0.0 -1.0 0.0
BOUN V VALU POWE Y Y+ 10.0 0.0 -1.0 0.0
/----------------------------------------------------------------------/
/ Fluid Properties and Constants
DENSity 1.0
VISC 1.0
SPEC 1.0
PRAN EFFE 0.7
/----------------------------------------------------------------------/
/ SOLUTION OPTIONS
DIFF SECO SKEW U V P
/
DIAGnostic output U V P RP at (6,6) every 50 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (4,4)
OUTPut for SELEcted window
SAVE OFF U V P on file 'V11.SAV'
/----------------------------------------------------------------------/
/ OPERATIONAL CONTROL
CONV GLOB 1.E-10 1
SOLVE U V P STEADY 2000
/
END
/

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************************************************************************
TITLE Problem V.12 - AXIAL FLOW >>> C55
***** W (AXIAL VEL) = A * R + B
************************************************************************
/
GRID NODEs 22 by 22
COOR CORN (1,1) (2,1) (1,2) (2,2) CYLI
/----------------------------------------------------------------------/
/ Initial and Boundary Conditions
BOUN U VALU LINE Y X- 0.0 10.0
BOUN U VALU LINE Y X+ 0.0 10.0
BOUN U VALU LINE Y Y- 0.0 10.0
BOUN U VALU LINE Y Y+ 0.0 10.0
/----------------------------------------------------------------------/
/ Fluid Properties and Constants
DENSity 1.0
VISC 1.0
SPEC 1.0
PRAN EFFE 0.7
/----------------------------------------------------------------------/
/ SOLUTION OPTIONS
DIFF SECO SKEW U V P
/
DIAGnostic output U V P RP at (6,6) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (2,2)
OUTPut for SELEcted window
SAVE OFF U V P on file 'V12.SAV'
/----------------------------------------------------------------------/
/ OPERATIONAL CONTROL
CONV GLOB 1.E-10 1
SOLVE U V P STEADY 1000
/
END
/

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************************************************************************
TITLE Problem V.13 - AXIAL POISEUELLE FLOW >>> C56
***** U = A * R**2
************************************************************************
/
GRID NODEs 42 by 42 !!! <==== Changed by akr from 62 by 62
COOR CORN (1,1) (2,1) (1,2) (2,2) CYLI
/----------------------------------------------------------------------/
/ Initial and Boundary Conditions
BOUN U VALU POWE Y X- 10.0 0.0 2.0 0.0
BOUN U VALU POWE Y X+ 10.0 0.0 2.0 0.0
BOUN U VALU POWE Y Y- 10.0 0.0 2.0 0.0
BOUN U VALU POWE Y Y+ 10.0 0.0 2.0 0.0
/----------------------------------------------------------------------/
/ Fluid Properties and Constants
DENSity 1.0
VISC 1.0
SPEC 1.0
PRAN EFFE 0.7
/----------------------------------------------------------------------/
/ Source and Sink Specifications
SOUR U CONS VOLU -40.0
/----------------------------------------------------------------------/
/ SOLUTION OPTIONS
DIFF SECO SKEW U V P
/
DIAGnostic output U V P RU RV at (6,6) every 50 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (4,4)
OUTPut for SELEcted window
SAVE OFF U V P on file 'V13.SAV'
/----------------------------------------------------------------------/
/ OPERATIONAL CONTROL
CONV GLOB 1.E-10 1
SOLVE U V P STEADY 2000
/
END
/

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************************************************************************
TITLE Problem V.14 - LAMINAR FULLY DEVELOPED FLOW IN A PIPE >>> C57
***** u_cl = (source of u)*(r**2)/(4*visc)
***** where u_cl=center line velocity and r is pipe radius.
************************************************************************
/
GRID NODEs 3 by 42
COORdinate corners (0.,0.) (1.,0.) (0,1.) (1.,1.) CYLI
/
/ Initial and Boundary Conditions
SYMM at Y-
WALL at Y+
PERIodic in X direction
/
/ Fluid Properties and Constants
DENSity 1.0
VISC 1.0
SPEC 1.0
/
SOUR U CONS VOLU 1000.0
RELAX U=1.0, V=1.0
/
DIAGnostic output U P at (2,21) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (999,999) interval (1,3)
OUTPut U for SELEcted window
SAVE OFF U V P on file 'V14.SAV'
/----------------------------------------------------------------------/
/ OPERATIONAL CONTROL
CONV U REFE 1.E-6
SOLVE U P STEADY 100
/
END
/

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************************************************************************
TITLE Problem V.15: CONDIF TEST CASE FOR STEP FLOW - theta = 45 deg
***** A.K. Runchal (1987), Int. J. Num. M. Eng., 24, p.1593
///// This problem can be set up in two ways:
///// Case A: Grid aligned with x & y; velocity at an angle
///// Case B: Grid at an angle but velocity aligned with x or y
************************************************************************
/
GRID NODEs 12 by 12
/
/COORdinate X RANGe 1. !! Case A
/COORdinate Y RANGe 1. !! Case A
/SET U = 0.70710678 !! Case A
/SET V = 0.70710678 !! Case A
/
COORdinates: 0. -0.70710678 +0.70710678 0. !! Case B
-0.70710678, 0. 0. +0.70710678 !! Case B
SET U = 0 !! Case B
SET V = 1 !! Case B
/
BOUNdary X- : T = 1
BOUNdary Y- : T = 0.
BOUNdary X+ : T FLUX = 0.
BOUNdary Y+ : T FLUX = 0.
/
DENSity 1.0
VISCosity 0
SPECific heat 1.0
PRANdtl number EFFEctive 1.0
RELAxation factor: T = 1
/INTEGRATION BY CONDIF FACTOR = 4
/
DIAGnostic T output at (7,7) every 20 steps
DEBUG GEOMERTY OFF
FLUX DEFAult output OFF
SELEct from (1,1) to (11,999) interval (1,2)
OUTPut T for SELEcted window
SAVE OFF T 'V15.SAV'
/
SOLVE T for 500 200 steps in STEAdy mode
/
END
/

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