FUNDAMENTALS OF GAS-PARTICLE FLOW
SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS…… 1 1 FLOW OVER PLATES, VELOCITY PROFILES, SHEAR FORCES, HEAT TRANSFER 1 1 1 LAMINAR FLOW OVER ONE SITE OF A PLANE 1 1 2 TURBULENT FLOW PARALLEL TO PLANE 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS 1 2 1 HYDRAULICALLY SMOOTH WALL SURFACE 1 2 2 TRANSITION REGION 1 2 3 COMPLETELY ROUGH REGION 1 2 4 HEAT TRANSFER TO FLUID IN A PIPE 1 3 TRANSIENT FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS NOMENCLATURE REFERENCES INTRODUCTION TO TURBULENCE OF MULTI-PHASE FLOWS 2 1 BASIC IDEAS 2 2 ISOTROPY 2 3 SCALES, EDDY VISCOSITY 2 3 1 SMALL SCALE TURBULENT MOTION 2 3 2 LARGE SCALE TURBULENT MOTION, KOLMOGOROV-PANDTL EXPRESSION 2 4 K-EPS FRAMEWORK SOURCES FOR FINE RESOLUTION OUTSIDE THE BOUNDARY LAYER 3 1 BULK SOURCES 3 1 1 DEFORMATION OF THE VELOCITY FIELD 3 1 2 BLOWING AND SUCTION 3 1 3 BUOYANCY DRIVEN TURBULENCE GENERATION 3 1 4 TURBULENCE GENERATED IN PARTICLE TRACES 3 2 TURBULENCE GENERATION DUE TO NUCLEATE BOILING 3 3 TREATMENT OF THE BOUNDARY LAYER FOR NON-BOILING FLOWS 3 4 INITIAL CONDITIONS SOURCE TERMS FOR K-EPS MODELS IN POROUS STRUCTURES 4 1 SINGLE-PHASE FLOW 4 1 1 STEADY DEVELOPED GENERATION DUE TO WALL FRICTION 4 1 2 HEAT TRANSFER AT THE WALL FOR STEADY DEVELOPED FLOW 4 1 3 HEAT TRANSFER AT THE WALL FOR NON-DEVELOPED OR TRANSIENT FLOW 4 1 4 SINGULARITIES 4 2 MULTIPHASE FLOW 4 2 1 STEADY DEVELOPED GENERATION DUE TO WALL FRICTION 4 2 2 HEAT TRANSFER AT THE WALL FOR FORCED CONVECTION WITHOUT BOILING 4 2 3 CONTINUUM-CONTINUUM INTERACTION 4 2 4 SINGULARITIES 4 2 5 DROPLETS DEPOSITION AT WALLS FOR STEADY DEVELOPED FLOW 4 2 6 DROPLETS DEPOSITION AT WALLS FOR TRANSIENT FLOW INFLUENCE OF THE INTERFACIAL FORCES ON THE TURBULENCE STRUCTURE 5 1 DRAG FORCES 5 2 THE ROLE OF THE LIFT FORCE IN TURBULENT FLOWS 5 3 LUBRICATION FORCE IN THE WALL BOUNDARY LAYER 5 4 THE ROLE OF THE DISPERSION FORCE IN TURBULENT FLOWS 5 4 1 DISPERSED PHASE IN LAMINAR CONTINUUM 5 4 2 DISPERSED PHASE IN TURBULENT CONTINUUM PARTICLE-EDDY INTERACTIONS 6 1 THREE POPULAR MODELLING TECHNIQUES 6 2 PARTICLE-EDDY INTERACTION WITHOUT COLLISIONS 6 2 1 RESPONSE COEFFICIENT FOR SINGLE PARTICLE 6 2 2 RESPONSE COEFFICIENT FOR CLOUDS OF PARTICLES 6 2 3 PARTICLE-EDDY INTERACTION TIME WITHOUT COLLISIONS 6 3 PARTICLE-EDDY INTERACTION WITH COLLISIONS TWO GROUP K-EPS MODELS 7 1 SINGLE PHASE FLOW 7 2 TWO-PHASE FLOW SET OF BENCHMARKS FOR VERIFICATION OF K-EPS MODELS IN SYSTEM COMPUTER CODES 8 1 INTRODUCTION 8 2 SINGLE PHASE CASES 8 3 TWO-PHASE CASES CONCLUSIONS SIMPLE ALGEBRAIC MODELS FOR EDDY VISCOSITY IN BUBBLY FLOW 9 1 SINGLE-PHASE FLOW IN ROD BUNDLES 9 1 1 PULSATIONS NORMAL TO THE WALL 9 1 2 PULSATION THROUGH THE GAP 9 1 3 PULSATION PARALLEL TO THE WALL 9 2 TWO-PHASE FLOW 9 2 1 SIMPLE ALGEBRAIC MODELS 9 2 2 LOCAL ALGEBRAIC MODELS IN THE FRAMEWORK OF THE BOUSSINESQ’S HYPOTHESIS 9 2 3 MODIFICATION OF THE BOUNDARY LAYER SHEAR DUE TO MODIFICATION OF THE BULK TURBULENCE 9 2 4 MODIFICATION OF THE BOUNDARY LAYER SHEAR DUE BOILING AT THE WALL LARGE EDDY SIMULATION 10 1 PHENOMENOLOGY 10 2 FILTERING – BRIEF INTRODUCTION 10 3 THE EXTENSION OF THE AMSDEN ET AL LES MODEL TO POROUS STRUCTURES SOLUBILITY OF O2, N2, H2 AND CO2 IN WATER 11 1 INTRODUCTION 11 2 OXYGEN IN WATER 11 3 NITROGEN WATER 11 4 HYDROGEN WATER 11 5 CARBON DIOXIDE–WATER 11 6 DIFFUSION COEFFICIENTS 11 7 EQUILIBRIUM SOLUTION AND DISSOLUTION TRANSIENT SOLUTION AND DISSOLUTION OF GASSES IN LIQUID FLOWS 12 1 BUBBLES 12 1 1 EXISTENCE OF MICRO-BUBBLES IN WATER 12 1 2 HETEROGENEOUS NUCLEATION AT WALLS 12 1 3 STEADY DIFFUSION MASS TRANSFER OF THE SOLVENT ACROSS BUBBLE INTERFACE 12 1 4 INITIAL BUBBLE GROWTH IN WALL BOUNDARY LAYER 12 1 5 TRANSIENT DIFFUSION MASS TRANSFER OF THE SOLVENT ACROSS THE BUBBLE INTERFACE 12 2 DROPLETS 12 2 1 STEADY STATE GAS SITE DIFFUSION 12 2 2 TRANSIENT DIFFUSION INSIDE THE DROPLET 12 3 FILMS 12 3 1 GEOMETRICAL FILM-GAS CHARACTERISTICS 12 3 2 LIQUID SIDE MASS TRANSFER DUE TO MOLECULAR DIFFUSION 12 3 3 LIQUID SIDE MASS TRANSFER DUE TO TURBULENCE DIFFUSION THERMODYNAMIC AND TRANSPORT PROPERTIES OF DIESEL FUEL… 13 1 INTRODUCTION 13 2 CONSTITUENTS OF DIESEL FUEL 13 3 AVERAGED BOILING POINT AT ATMOSPHERIC PRESSURE 13 4 REFERENCE LIQUID DENSITY POINT 13 5 CRITICAL TEMPERATURE, CRITICAL PRESSURE 13 6 MOLAR WEIGHT, GAS CONSTANT 13 7 SATURATION LINE 13 8 LATENT HEAT OF EVAPORATION 13 9 THE LIQUID DENSITY 13 9 1 THE VOLUMETRIC THERMAL EXPANSION COEFFICIENT 13 9 2 ISOTHERMAL COEFFICIENT OF COMPRESSIBILITY 13 10 LIQUID VELOCITY OF SOUND 13 11 THE LIQUID SPECIFIC HEAT AT CONSTANT PRESSURE 13 12 SPECIFIC LIQUID ENTHALPY 13 13 SPECIFIC LIQUID ENTROPY 13 14 LIQUID SURFACE TENSION 13 15 THERMAL CONDUCTIVITY OF LIQUID DIESEL FUEL 13 16 CINEMATIC VISCOSITY OF LIQUID DIESEL FUEL 13 17 DENSITY AS A FUNCTION OF TEMPERATURE AND PRESSURE FOR DIESEL FUEL VAPOR 13 18 SPECIFIC CAPACITY AT CONSTANT PRESSURE FOR DIESEL VAPOR 13 19 SPECIFIC ENTHALPY FOR DIESEL FUEL VAPOR 13 20 SPECIFIC ENTROPY FOR DIESEL FUEL VAPOR 13 21 THERMAL CONDUCTIVITY OF DIESEL FUEL VAPOR 13 22 CINEMATIC VISCOSITY OF DIESEL FUEL VAPOR 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS HUNDREDS OF VERY USEFUL CONSTITUTIVE RELATIONS THAT DESCRIBE THE INTERACTIONS IN MULTIPHASE FLOWS ARE BASED ON THE ACHIEVEMENTS OF SINGLE-PHASE BOUNDARY LAYER THEORY THAT IS WHY IT IS IMPORTANT TO RECALL AT LEAST SOME OF THEM, BEFORE MOVING ON TO MORE COMPLEX INTERACTIONS IN MULTIPHASE FLOW THEORY MY FAVORITE BOOK TO START LEARNING THE MAIN IDEAS OF SINGLE-PHASE BOUNDARY LAYER THEORY IS THE FAMOUS MONOGRAPH BY SCHLICHTING (1982) THIS CHAPTER GIVES ONLY THE BASICS REQUIRED TO UNDERSTAND THE REST OF THE BOOK 1 1 FLOW OVER PLATES, VELOCITY PROFILES, SHEAR FORCES, HEAT TRANSFER 1 1 FLOW PROFILES OVER PLATES, VELOCITY, SHEAR FORCES CONSIDER CONTINUUM FLOW PARALLEL TO A PLATE ALONG THE X-AXIS HAVING VELOCITY FAR FROM THE SURFACE EQUAL TO U? THE SHEAR FORCE ACTING ON THE SURFACE PER UNIT FLOW VOLUME IS THEN W W W FLOW F F V ? = , (1 1) WHERE THE WALL SHEAR STRESS IS USUALLY EXPRESSED AS ( ) W W C X U ? ? ? = (1 2) HERE THE FRICTION COEFFICIENT W C IS OBTAINED FROM THE SOLUTION OF THE MASS AND MO- MENTUM CONSERVATION AT THE SURFACE 1 1 1 LAMINAR FLOW OVER ONE SITE OF A PLANE FOR LAMINAR FLOW OVER ONE SITE OF A PLANE, THE SOLUTION OF THE MOMENTUM EQUATION DELIVERS THE LOCAL SHEAR STRESS AS A FUNCTION OF THE MAIN FLOW VELOCITY AND OF THE DISTANCE FROM THE BEGINNING OF THE PLATE AS FOLLOWS 2 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS ( ) ( ) 1/ W W X C X U X U ? ? ? ? ? = = ? ? ? ? ? ? , (1 3) EQ (7 32) SCHLICHTING (1982) P 140 THE AVERAGED DRAG COEFFICIENT OVER X ? IS THEN ( ) , 1/ W W X F Y X C U X U ? ? ? ? ? ? ? = = ? ? ? ? ? ? ? , RE 5 X ? < × , (1 4) EQ (7 34) SCHLICHTING (1982) P 141 THE CORRESPONDING HEAT TRANSFER COEFFICIENTS H ARE REPORTED TO BE 1/ PR X U X HX NU ? ? ? ? ? ? = = ? ? ? ? FOR PR ? FOR LIQUID METALS, (1 5) 1/ < < , (1 6) 1/ PR X U X HX NU ? ? ? ? ? = = ? ? ? ? FOR PR ??, (1 7) SCHLICHTING (1982) P 303 AVERAGING OVER X ? RESULTS IN X X NU NU ? ? = (1 8) NOTE THAT JUKAUSKAS AND JYUGJA (1969) REPORTED THE SAME RESULT: ( ) 1/ PR PR PR X W U X HX NU ? ? ? ? ? = = ? ? ? ? , (1 9) ALSO TAKING INTO ACCOUNT THE WALL INFLUENCE ( PRW IS COMPUTED AT WALL TEMPERATURE) MIHEEV AND MIHEEVA (1973) REPORTED A FAVORABLE COMPARISON WITH THE ABOVE COR- RELATION WITH EXPERIMENTAL DATA IN THEIR FIG 3-6, P 69 1 1 FLOW PROFILES OVER PLATES, VELOCITY, SHEAR FORCES W W X C X U X U ? ? ? ? ? = = ? ? ? ? ? ? , (1 10) EQ (21 12) SCHLICHTING (1982) P 653 THIS EQUATION IS OBTAINED BY ASSUMING THE VALIDITY OF THE SO-CALLED 1/7-TH VELOCITY PROFILE, ( ) ( ) 1/ MAX , U X Y Y U X ? ? ? = ? ? ? ? ? ? , (1 11) WITH THE BOUNDARY LAYER THICKNESS VARYING WITH THE DISTANCE FROM THE EDGE OF THE PLATE IN ACCORDANCE WITH ( ) 1/ U X X X ? ? ? ? ? = ? ? ? ? (1 12) AT A DISTANCE FROM THE WALL GIVEN BY 99% 5 U X Y ? ? ? = ? (1 13) THE VELOCITY REACHES 99% OF THE FLOW MEAN VELOCITY 99% ? IS CALLED THE DISPLACE- MENT THICKNESS THE AVERAGED STEADY STATE DRAG COEFFICIENT OVER X ? IS THEN ( ) , 1/ W W X X F Y X C U ? ? ? ? ? ? = = , PR X U X H X NU ? ? ? ? ? ? ? ? = = ? ? ? ? , (1 16) RESPECTIVELY THE INFLUENCE OF THE WALL PROPERTIES IN THE LAST EQUATION IS PROPOSED BY KNUDSEN AND KATZ (1958) TO BE TAKEN INTO ACCOUNT BY COMPUTING THE PROPERTIES AT THE FOLLOWING EFFECTIVE TEMPERATURE ( ) 0 1PR EFF W T T T T + = + ? + (1 17) THE ONLY KNOWN INFORMATION FOR THE INFLUENCE OF THE UNSTEADINESS OF THE FAR FIELD VELOCITY IS GIVEN BY SIDOROV (1959), , 1/ PR PR PR X W U X H X NU ? ? ? ? ? ? ? ? = = ? ? ? ? , (1 20) TAKING ALSO INTO ACCOUNT THE WALL INFLUENCE ( PRW IS COMPUTED AT WALL TEMPERATURE) MIHEEV AND MIHEEVA (1973) REPORTED A FAVORABLE COMPARISON WITH THE ABOVE COR- RELATION WITH EXPERIMENTAL DATA IN THEIR FIG 3-7, P 70 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS CONSIDER CONTINUUM FLOW ALONG THE X-AXIS OF A CIRCULAR PIPE HAVING VELOCITY CROSS SECTION AVERAGED VELOCITY EQUAL TO W THE SHEAR FORCE ACTING ON THE SURFACE PER UNIT FLOW VOLUME IS THEN W W W FLOW F F V ? = , (1 21) WHERE THE WALL SHEAR STRESS IS USUALLY EXPRESSED AS 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS W W C W ? ? = (1 22) HERE THE FRICTION COEFFICIENT W C , CALLED THE FANNING FACTOR IN THE LITERATURE, IS OB- TAINED FROM THE SOLUTION OF THE DEVELOPED STEADY STATE MASS AND MOMENTUM CON- SERVATION IN THE PIPE REPLACING THE WALL SURFACE TO PIPE VOLUME RATIO WITH H D , WE HAVE FOR THE WALL FRICTION FORCE PER UNIT VOLUME OF THE FLOW FR W W W W FLOW H H F F C W C W W V D D ? ? ? ? = = = (1 23) WHERE FR W C ? = , (1 24) CALLED THE FRICTION COEFFICIENT, IS USUALLY USED IN EUROPE NOTE THE FACTOR OF 4 BE- TWEEN THE FANNING FACTOR AND THE FRICTION COEFFICIENT AND FR W W ? ? ? = (1 25) NOTE THAT FOR A STEADY DEVELOPED SINGLE FLOW, THE MOMENTUM EQUATION READS ( ) D DP R R DR DX ? ? = WITH W DP DP DX DX = , AND THEREFORE ( ) W DP D R R DR DX ? ? = , WE HAVE ( ) W DP D R RDR DX ? = , OR AFTER INTEGRATING ( ) W DP R R DX ? = , (1 26) WHICH GIVES AT THE WALL THE RELATION BETWEEN THE WALL SHEAR STRESS AND THE PRESSURE GRADIENT DUE TO FRICTION W W DP R DX ? = (1 27) USUALLY FOR DESCRIBING TURBULENT FLOWS IN PIPE THE FOLLOWING DIMENSIONLESS VARI- ABLES ARE USED: THE FRICTION VELOCITY 6 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS * FR W W W ? ? ? = = , (1 28) THE DIMENSIONLESS CROSS SECTION AVERAGED VELOCITY * W W W + = , (1 29) AND THE DIMENSIONLESS DISTANCE FROM THE WALL * Y Y W ? + = , (1 30) WHERE Y IS THE DISTANCE FROM THE WALL NOTE THAT * Y W ? IS IN FACT THE DEFINITION OF A BOUNDARY LAYER REYNOLDS NUMBER WITH THIS TRANSFORMATION THE MEASURED MEAN VELOCITY DISTRIBUTION NEAR THE WALL IS NOT STRONGLY DEPENDENT ON THE REYNOLDS NUM- BER AS SHOWN IN FIG 1 1 HAMMOND (1985) APPROXIMATED THIS DEPENDENCY BY A CONTINUOUS FUNCTION OF THE TYPE ( ) + + + = Y Y U , WHICH MUST BE INVERTED ITERATIVELY IF ONE NEEDS ( ) + + + = U U Y U + Y + FIG 1 1 MEAN VELOCITY DISTRIBUTION NEAR THE WALL, LAUFER J (1953) THE PENETRATION OF THE WALL ROUGHNESS K INTO THE BOUNDARY LAYER DICTATES DIFFERENT SOLUTIONS OF PRACTICAL INTEREST USUALLY THE DIMENSIONLESS ROUGHNESS OF THE SURFACE * K KW ? + = IS COMPARED TO THE CHARACTERISTIC DIMENSIONLESS SIZES OF THE BOUNDARY LAYER TO DEFINE THE VALIDITY REGION OF THE SPECIFIC SOLUTION OF THE MOMENTUM EQUATION 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS 1 2 1 HYDRAULICALLY SMOOTH WALL SURFACE HYDRAULICALLY SMOOTH SURFACES ARE DEFINED IF K + ? ? (1 31) 1 2 1 1 THE BLASIUS SOLUTION IN 1911, BLASIUS OBTAINED THE FOLLOWING EQUATION, 1/ 0 3164 RE FR ? = , (1 32) WHERE RE H WD ? = THIS RESULT HAS BEEN VALIDATED WITH HIS DATA AND THE DATA OF OTHER AUTHORS FOR < LATER IT WAS FOUND THAT THE VELOCITY PROFILE ASSOCIATED WITH THIS FRICTION COEFFICIENT HAS THE FORM ( ) ( )( ) 1/ 1/ MAX N N N N Y Y W Y W W R R N + + ? ? ? ? = = ? ? ? ? ? ? ? ? (1 33) IT IS KNOWN FROM THE NIKURADSE MEASUREMENTS THAT THE EXPONENT IS A FUNCTION OF REYNOLDS NUMBER: FOR RE 1 1 N = ( ) 1/ MAX Y Y W Y W W R R ? ? ? ? = = ? ? ? ? ? ? ? ? , (1 34) OR ( ) ( ) 1/ W Y Y + + = (1 35) AND FOR RE 3 2 N = 1 2 1 2 THE COLLINS ET AL SOLUTION A MORE SOPHISTICATED SOLUTION THAN THE BLASIUS PROFILE, WHICH DEPENDS ON THE REY- NOLDS NUMBER, WAS PROPOSED BY COLLINS ET AL (1978) AND BENDIKSEN (1985): ( ) ( ) ? = + ? ? ? ? ?, (1 37) 8 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS ( ) N ? ? ? ? ? ? = ? ? ? ? ? ? + ? ? (1 38) THE RESULTING FRICTION COEFFICIENT IS THEN FR ? = ? (1 39) 1 2 1 3 THE VON KARMAN UNIVERSAL VELOCITY PROFILES VELOCITY PROFILE: A MORE ACCURATE MATHEMATICAL REPRESENTATION OF THE VELOCITY PRO- FILES IN FIG 1 1 WAS GENERALIZED BY VON KARMAN (1939), USING THE PRANDTL MIXING LENGTH THEORY W DW W DY ? ?A , U DW U DY ? ?A , WU WU W U DW DW C W U C DY DY ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? A A A (1 40) HERE WU W U C W U ? ? = ? ? (1 41) IS CALLED CORRELATION COEFFICIENTS BETWEEN THE FLUCTUATIONS IN BOTH DIRECTIONS THE EFFECTIVE TURBULENT CINEMATIC VISCOSITY IS ASSUMED TO BE PROPORTIONAL TO THE VELOC- ITY GRADIENT ? = IS CALLED THE VON KARMAN CONSTANT THE SHEAR STRESS IN THE BOUNDARY LAYER IS THEN WU DW DW DW DW W U C W U DY DY DY DY ? ?? ? ?? ? ?? ? ? ? ? ? ? ? = + = + = + ? ? ? ? A (1 44) THE DIMENSIONLESS SHEAR STRESS IS DIVIDED BY THE WALL SHEAR STRESS TO GIVE 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS W W W DW DW DW DW DY DY DY DY ? ? ? ? ? ? ? + + + + + ? ? ? ? = + = + ? ? ? ? ? ? ? ? A A (1 45) FOR A BOUNDARY LAYER FLOW WITH ZERO PRESSURE GRADIENT Y ? ? ?? AT THE WALL AND THEREFORE W ? ? ? , THE QUADRATIC EQUATION DW DW DY DY + + + + + ? ? + ?= ? ? ? ? A (1 46) CAN BE SOLVED WITH RESPECT TO THE GRADIENT AND INTEGRATED OVER Y IN THIS WAY A VE- LOCITY PROFILE CAN BE GENERATED FOR NEGLIGIBLE VISCOUS STRESS DW DY + + + = A FOR MIXING LENGTH Y ? + + = A , THE EQUATION LN DW D Y ? + + = HAS THE ANALYTICAL SOLUTION 1 LN W Y CONST ? + + = + , (1 47) WHICH IS NOT DEPENDENT ON THE MOLECULAR VISCOSITY THIS IS THE VON KARMAN LAW FOR FULLY TURBULENT FLOW SCHLICHTING FOUND THAT THE DATA OF NIKURADSE (1933) FOR RE 3 4 W W W = ? (1 49) FRICTION COEFFICIENT: THE ABOVE PROFILE DICTATES THE FOLLOWING EXPRESSION FOR THE FRICTION FACTOR ( ) FR FR ? ? = ? , (1 50) SCHLICHTING (1982) P 624 IN FACT THIS IS EQ (43) FROM VAN DRIEST (1955) P 1011 KNOWING THE FRICTION FACTOR THE FRICTION VELOCITY IS * FR W W ? = (1 51) AND THE PROFILE IS DEFINED 10 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS VON KARMAN (1939) INTRODUCED A BUFFER ZONE BETWEEN THE VISCOUS AND FULLY TURBULENT LAYERS ALREADY INTRODUCED BY PRANDTL (1910) IN THE FIRST ZONE CLOSE TO THE WALL THE FLOW IS LAMINAR IN THE SECOND AND THIRD THE CONSTANTS ARE COMPUTED SO THAT THEY HAVE SMOOTH PROFILES TABLE 1 1 SUMMARIZES THE SCHLICHTING VELOCITY PROFILES TABLE 1 1 VELOCITY PROFILES IN THE BOUNDARY LAYER SUB-LAYER DEFINED BY VELOCITY PROFILE VISCOUS Y+ ? W Y + + = BUFFER W Y + + = ? FULLY TUR- BULENT W Y + + = + TABLE 1 2 CONTAINS SOME IMPORTANT INTEGRALS OF THE WIDELY USED UNIVERSAL PROFILE + ? IS THE VOLUMETRIC FLOW RATE PER UNIT WIDTH OF THE WALL TABLE 1 2 DIMENSIONLESS VOLUMETRIC FLOW PER UNIT WIDTH OF THE SURFACE SUB-LAYER W DY ? + + + + ? = ? DW DY DY ? + + + + ? ? ? ? ? ? ? VISCOUS ( ) ? + ? ? + BUFFER ? ? ? + + + ? + ? ? ? ? + + + + ? 6 25 ? + ?? NOTE THAT THE BOUNDARY LAYER REYNOLDS NUMBER, DEFINED AS * RE 0 707RE? ? + = FOR 0 482RE? ? + = FOR ? < ? , (1 54) 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS 0 095RE? ? + = FOR 0 7071RE ? + = FOR 0 0504RE ? + = FOR RE < , (1 57) WHERE ( ) RE H W D ? ? = , H D IS THE PIPE DIAMETER AND THE SUBSCRIPT 2 INDICATES A FILM OBSERVE THE DIFFERENCES IN CONSTRUCTING THE REYNOLDS NUMBERS RE? AND RE : IN THE FIRST CASE THE CHARACTERISTIC LENGTH IS 4? , WHILE IN THE SECOND IT IS H D AFTER PRANDTL AND VON KARMAN MORE COMPLICATED EXPRESSIONS FOR THE MIXING LENGTH WERE PROPOSED: VAN DRIEST (1955) INTRODUCED THE SO-CALLED DAMPING FUNC- TION ( ) 1 EXP / Y Y ? + ? ? = ? ? ? ? A , (1 58) WHICH REFLECTS THE FACT THAT THE FLUCTUATIONS ARE DIMINISHING CLOSE TO THE WALL, AN OBSERVATION ALREADY MADE BY STOKES (1845) IT IS KNOWN THAT THE CONSTANT ACTUALLY DEPENDS ON THE REYNOLDS NUMBER AND TAKES VALUES BETWEEN 20 AND 30 THE ADVAN- TAGE OF THIS FUNCTION IS THAT IT GIVES A SMOOTH VELOCITY PROFILE OVER THE THREE LAYERS DISCUSSED ABOVE VAN DRIEST MADE THE OBSERVATION THAT ROUGHNESS INTRODUCES ADDITIONAL TURBU- LENCE IN THE BOUNDARY LAYER, SO THAT FOR K + > NO DAMPING IS EXPECTED FORMALLY IT IS EXPRESSED BY ( ) 1 EXP EXP Y Y Y K ? + + + ? ? ? ? ? ? = ? ? + ? ? ? ? ? ? ? ? ? ? ? A FOR K + < , (1 59) WHICH GOES TO EQ (1 58) IF THE ROUGHNESS GOES TO ZERO FOR ROUGH WALLS DEFINED WITH K + > THERE IS NO LONGER ANY DAMPING, AND Y ? = A FOR PIPE FLOW, GRÖTZBACH (2007) RECOMMENDED COMBINING THE NIKURADSE (1932) MIXING LENGTH PARABOLA WITH THE VAN DRIEST DAMPING FACTOR: ( ) 0 08 0 06 1 EXP Y Y Y R R R + ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A (1 60) 12 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS HETEROGENEOUS TURBULENCE IN THE BOUNDARY LAYER: IT WAS EXPERIMENTALLY OBSERVED BY LAUFER J (1952, 1953) THAT THE FLUCTUATIONS OF THE VELOCITY ARE EQUILATERAL IN THE CENTRAL PART OF THE PIPE BUT HETEROGENEOUS CLOSE TO THE WALL (FIG 1 2) THIS IS CON- FIRMED BY MANY AUTHORS E G QUARMBY AND QUIRK (1974) LAUFER’S DATA INDICATE THAT THE FLUCTUATIONS OF THE AXIAL VELOCITIES NEAR THE WALL REGION ARE ABOUT THREE TIMES LARGER THAN THE FLUCTUATIONS IN THE OTHER DIRECTIONS – HETEROGENEOUS TURBU- LENCE THE RADIAL FLUCTUATION VELOCITY CAN BE APPROXIMATED BY A BOLTZMANN FUNC- TION ( ) Y Y DY A A U A E + + + + ? ? ? = + + , (1 61) WHERE 1A = –30 333 65, 2A = 0 89475, 0Y+ = –43 514 54 AND DY+ = 12 72364 10 20 30 40 50 60 70 80 AXIAL RE RADIAL ANGULAR V' + Y + FIG 1 2 VELOCITY FLUCTUATIONS AS A FUNCTION OF THE DISTANCE FROM THE WALL MEASURED BY LAU- FER J (1953) OBSERVE THAT THE DIMENSIONLESS RADIAL FLUCTUATION VELOCITY IS ALMOST INDEPENDENT OF THE REYNOLDS NUMBER AND THAT FOR * T W D ? = (NOTE THAT 0 0414 * T E V W D ? ? ? = ? = ) KNOW- ING THE FLUCTUATION OF THE NORMAL TO THE WALL VELOCITY IN A PIPE FLOW IS IMPORTANT FOR ANALYZING DEPOSITION PROCESSES IN PARTICLE-LOADED FLOWS THIS IS ALSO ESSENTIAL FOR POST-CRITICAL HEAT TRANSFER DESCRIPTION IN ANNULAR FLOW WITH DROPLETS IN THE GAS CORE 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS U Y W Y + + ? = + , (1 63) V W R R Y ? ? = ? ? ? ? A (1 66) JOHANSEN (1991) REPORTED THE FOLLOWING APPROXIMATION FOR PIPE AND CHANNELS FLOWS WITH RADIUS OR HALF WITH R VALID NOT ONLY IN THE BOUNDARY LAYER BUT IN THE EN- TIRE CROSS SECTION: A) THE EDDY VISCOSITY T Y ? ? + ? ? = ? ? ? ? , Y+ < , (1 67) T Y ? ? + ? ? = ? ? ? ? ? , Y+ ? , (1 69) 14 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS IS IN AGREEMENT WITH THE PROFILES COMPUTED WITH DIRECT NUMERICAL SIMULA- TION REPORTED BY KIM ET AL (1987) B) THE PROFILE OF THE TIME-AVERAGED AXIAL VELOCITY IN ACCORDANCE WITH THE ABOVE EXPRESSIONS IS ( ) W Y Y W + ?? ? = ? ? ? ? , Y Y + + ? , (1 70) ( ) 2 5LN * W Y Y W Y + + ? ? + = + ? ? + ? ? , Y Y + + > , (1 71) Y V + + ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? , Y+ ? , (1 75) ( ) ( ) Y V V V R + + + + + ? ? ? ? ? ? = ? ? ? ? ? , Y R + + ? ? , (1 76) AGREE WITH THE MEASUREMENTS OF KUTATELADZE ET AL (1979) THE CHARACTER- ISTIC TIME SCALE OF THE FLUCTUATION IS THEN GIVEN BY T E V ? ? ? ? = WALL BOUNDARY CONDITIONS FOR 3D-MODELING: USING K ? ? MODELS IN COMPUTER CODES WITH LARGE SCALE DISCRETIZATION IS VERY POPULAR NOWADAYS IN THESE CODES THE BOUNDARY LAYER CANNOT BE RESOLVED FOR COMPUTING THE BULK CHARACTERISTICS BOUND- ARY CONDITIONS AT THE WALL ARE REQUIRED USUALLY A POINT CLOSE TO THE WALL E G W Y + + = + STARTS TO BECOME VALID AT THIS POINT THE VALUES OF THE TURBULENT KINETIC ENERGY PER UNIT MASS AND ITS DISSI- PATION ARE 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS C? = COMING FROM THE DEFINITION EQUATION OF THE TURBULENT CINEMATIC VIS- COSITY T C K ? ? ? = , SEE FOR INSTANCE LEE ET AL (1986) BRADSHAW (1967) FOUND EXPERIMENTALLY A USEFUL RELATIONSHIP W K CONST ? ? ? THAT CAN BE USED FOR THIS PUR- POSE AS APPROXIMATION HARSHA AND LEE (1970) PROVIDED EXTENSIVE MEASUREMENTS SHOWING THE CORRECTNESS OF THE RELATION 3 3 W K ? ? ? FOR WAKES AND JETS COMPUT- ING THE FANNING FACTOR IN 2 W W C W ? ? = BY USING APPROPRIATE CORRELATION WE CAN APPROXIMATE THE SPECIFIC TURBULENT ENERGY AT THE WALL REGION NOTE THAT ALSHAMANI (1978) REPORTED THAT THERE IS NO LINEARITY IN THE BOUNDARY LAYER W K W ? ? +? ? ? , WHERE W +? IS THE DIMENSIONLESS FLUCTUATION OF THE AXIAL VELOCITY FOR COMPLETENESS, THE YU ET AL (2001) VELOCITY PROFILE IN A CIRCULAR PIPE IS SHOWN BY DIRECT NUMERICAL SIMULATION TO BE ADEQUATE ALSO FOR LIQUID METAL FLOWS WITH VERY LOW PRANDTL NUMBER ( ) W R R R + + + + = + ? + , (1 79) WHERE * R R W ? + = 1 2 1 4 THE REICHARDT SOLUTION IN LOOKING FOR AN APPROPRIATE PIPE FLOW TURBULENCE DESCRIPTION THAT WAS VALID NOT ONLY UP TO THE BOUNDARY LAYER BUT UP TO THE AXIS OF THE PIPE, REICHARDT (1951) RE- PRODUCED THE AVAILABLE DATA FOR HIGH REYNOLDS NUMBERS IN THE FORM OF TURBULENT CI- NEMATIC VISCOSITY AS A FUNCTION OF THE DISTANCE FROM THE WALL * 3 T Y Y W R R R ? ? ? ?? ? ? ? ? ? = + ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ?? ? , (1 80) IN ( ) T DW DY ? ? ? ? = + , W Y R ? ? = ? (1 81) MAKING A REASONABLE APPROXIMATION REICHARDT SUCCEEDED IN OBTAINING A SINGLE EQ- UATION FOR THE VELOCITY PROFILE THAT COVERS ALL THE REGIONS FROM THE WALL TO THE AXIS: 16 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS ( ) ( ) 1 5 2 5LN Y W Y R Y W Y R + ? ? ? ? ? ? ? ? ? ? ? ? ? = + ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ( ) ( ) 7 8 1 EXP Y Y Y + + + ? ? + ? ? ? ? ? ? ? ? (1 82) FOR Y << R THE EXPRESSION SIMPLIFIES TO ( ) ( ) 2 5LN * W Y Y W + = + ( ) ( ) 7 8 1 EXP Y Y Y + + + ? ? + ? ? ? ? ? ? ? ? (1 83) THE DERIVATIVE ( ) ( ) ( ) 2* 7 8 EXP DW W Y Y Y DY Y ? + + + + ? ? ? ? = + ? + ? ? ? ? ? ? + ? ? , (1 84) IS VERY USEFUL FOR COMPUTATION OF THE LIFT FORCE ACTING ON SMALL PARTICLES IN THE BOUNDARY LAYER AT Y R = FOR UPWARD BUBBLY FLOW IT IS DIRECTED FROM THE WALL INTO THE BULK FLOW AND FOR DROPLETS FROM THE BOUNDARY LAYER TOWARD THE WALL NOTE THAT LEE ET AL (1986) REPORTED AN ALTERNATIVE FORM OF THE CINEMATIC VISCOS- ITY AS A FUNCTION OF THE DISTANCE FROM THE WALL T Y Y Y Y Y R R R ? ? + + ? ?? ? ? ? ? ? ? ? ? ? = ? + ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (1 85) 1 2 2 TRANSITION REGION THE TRANSITION FROM THE HYDRAULICALLY SMOOTH TO COMPLETELY ROUGH REGION IS DE- FINED BY K + ? ? (1 86) IN THIS CASE THE VELOCITY PROFILE IS ( ) ( ) MAX *2 5LN W W Y W R Y ? = , (1 87) 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS MAX 3 75 * W W W = ? (1 88) THE FRICTION COEFFICIENT CORRELATION PROPOSED BY COLEBROOK (1939), 2LOG RE FR K R ? ? ? = ? + ? ? ? ? (1 90) NIKURADSE (1933) PROPOSED TO DESCRIBE THE VELOCITY PROFILE CLOSE TO THE WALL FOR SAND ROUGHNESS AS FOLLOWS ( ) 1 LN W Y E ? + + = , (1 91) WHERE E = 9 025 FOR E K + = FOR K + > (1 95) 1 2 3 COMPLETELY ROUGH REGION THE COMPLETELY ROUGH REGION IS DEFINED BY W W R K = + , (1 98) ( ) MAX 3 75 * W W Y W ? = (1 99) THE FRICTION COEFFICIENT FOUND BY VON KARMAN IS FR R K ? ? ? = + ? ? ? ? (1 100) A SLIGHT CHANGE MADE BY SCHLICHTING FR R K ? ? ? = + ? ? ? ? , (1 101) GIVES THE BEST FIT TO THE NIKURADSE DATA WHITE (2006) PROPOSED THE FOLLOWING APPROXIMATION OF THE VELOCITY PROFILE WITH AN OFFSET THAT DEPENDED ON THE TYPE OF THE ROUGHNESS ( ) * LN * W Y YW B B W ? ? ? ? = + ?? ? ? ? ? , (1 102) WHERE * LN B F TRAV C ? FOR GAS INTERACTION WITH THE TRAVELING WAVES 1 2 4 HEAT TRANSFER TO FLUID IN A PIPE COMPARING THE MOMENTUM AND THE ENERGY CONSERVATION EQUATION WITH STEADY DE- VELOPED FLOW IN A PIPE 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS T W Y DW R DY ? ? ? ? ? ? ? = + ? ? ? ? , (1 104) ( ) T W P Q Y DT A A C R DY ? ?? ? ? ? = ? + ? ? ? ? , (1 105) WE IMMEDIATELY REALIZE THE SIMILARITY NOTE THAT IN THIS CASE T W P T W Q C A A DT DU ? ? ? ?? + = ? + (1 106) FOR NEGLIGIBLE MOLECULAR VISCOSITY AND CONDUCTIVITY COMPARED TO THEIR TURBULENT COUNTERPART VALID FOR GASES AND PR T T T PC ? ? ? = ?, (1 107) REYNOLDS COMES TO THE REMARKABLE RELATION ( ) ?? ? = W P FR Q WC T ? ? , (1 108) SAYING THE HEAT TRANSFER BETWEEN WALL AND BULK FLOW IS PROPORTIONAL TO THE FRICTION COEFFICIENT REYNOLDS’ PROPOSAL TO CONSIDER THE TURBULENT CONDUCTIVITY PROPORTIONAL TO THE EDDY VISCOSITY PR T P T T C CONST ? ? ? = ? , (1 109) WAS VERY FRUITFUL IN OBTAINING PRACTICAL CORRELATIONS FOR DESCRIBING THE HEAT TRANSFER IN PIPES THEREFORE, KNOWING THE DEPENDENCE OF THE EDDY DIFFUSIVITY ON THE DISTANCE FROM THE WALL ( ) T W Y Y DW DY R ? ? ? ? ? ? = ? ? ? ? ? ? , (1 110) THE TURBULENT THERMAL CONDUCTIVITY IS ALSO KNOWN AND EQ (1 105) CAN BE INTE- GRATED THE RESULT IS THE TEMPERATURE DISTRIBUTION AS A FUNCTION OF THE WALL DISTANCE AS AN EXAMPLE BELOW WE GIVE THE MARTINELLI SOLUTION THE HEAT FLUX AT THE WALL IS THEN EASILY COMPUTED BECAUSE THE TEMPERATURE GRADIENT AT THE WALL IS KNOWN THIS IS THE METHOD THAT LEAD MANY AUTHORS TO A DERIVATION OF THE CORRELATION FOR THE HEAT TRANSFER AT THE WALL – AN IMPORTANT METHOD FOR THE TECHNOLOGICAL ACHIEVEMENT OF 20 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS TURBULENCE THEORY THE ANALOGY IS CALLED THE REYNOLDS ANALOGY AN EXPRESSION FOR THE HEAT TRANSFER COEFFICIENT WAS GENERATED BY PRANDTL; A MORE ACCURATE ONE BY VON KARMAN IS VALID UP TO PR = 25 A USEFUL CORRELATION SET REVIEWED BY KIRILLOV (1985, 1988) IS GIVEN BELOW 1 2 4 1 THE MARTINELLI SOLUTION FOR TEMPERATURE PROFILE TEMPERATURE PROFILES FOR HEATED WALLS CORRESPONDING TO THE ABOVE-MENTIONED THREE-LAYERS THEORY WERE FOUND BY MARTINELLI (1974), WHO ASSUMED THE VALIDITY OF THE REYNOLDS ANALOGY MARTINELLI INTEGRATED THE ENERGY CONSERVATION EQUATION FOR EACH OF THE THREE REGIONS AND OBTAINED THE DIMENSIONLESS TEMPERATURE ( ) ( ) * P W W T Y U C T T Y Q ? + ?? = ? ? ? ? ? , (1 111) AS A FUNCTION OF THE DIMENSIONLESS DISTANCE FROM THE WALL THE SOLUTIONS ARE GIVEN BELOW: FOR THE VISCOUS SUB-LAYER, Y+ ? , PR PR T T Y + + = (1 112) FOR THE BUFFER LAYER, Y+ ? ? , 5 PR PR LN PR PR T T Y T + + ? ? ? ? ? ? ? ? = + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (1 113) FOR THE TURBULENT CORE, Y+ < , ( ) 5 PR PR LN 1 5PR PR LN T T Y T + + ? ? ? ? ? ? = + + + ? ? ? ? ? ? ? ? ? ? (1 114) THESE ARE EQS 24, 25 AND 26, RESPECTIVELY, OBTAINED BY MARTINELLI (1974) THE BEST COMPARISON WITH THE EXPERIMENTAL DATA WAS FOUND FOR THE TURBULENT PRANDTL NUMBER SET TO 1: ( ) PR T H R D ? ? (1 116) A MORE COMPLEX RELATION WAS PROPOSED BY AZER AND CHAO (1960) FOR 0 6 < PR < 15, 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS 1 57RE PR EXP PR 1 135RE EXP T Y R Y R ? ? ? ? ? + ? ? ? = ? ? + ? ? ? (1 117) THE MARTINELLI SOLUTION IS IMPORTANT FOR ANALYZING NUCLEATE BOILING HEAT TRANSFER AT HEATED WALLS BECAUSE IT GIVES THE TEMPERATURE PROFILE IN THE BOUNDARY LAYER WHERE THE BUBBLE GENERATION HAPPENS STABLE BUBBLE GROWTH IS OBTAINED IF THE TEMPERA- TURE AT T = FOR Y+ < (1 120) AND ( ) ( ) ( ) { } { } T T T T ? ? ? ? ? ? ? ? ? = + ? ? ? ? ? ,(1 121) FOR T Y + + = + FOR Y+ < (1 123) AND ( ) PR T Y+ ? ? = + ? ? ? ?FOR Y+ > , (1 124) 22 1 SOME SINGLE-PHASE BOUNDARY LAYER THEORY BASICS WHICH IS ALSO USEFUL FOR PRANDTL NUMBERS UP TO 64, KAYS (1994) NA AND HABIB (1973) PROPOSED TO USE THE MODIFIED NIKURADZE PARABOLA WITH THE VAN DRIEST (1955) DAMPING FUNCTION ( ) MAX MAX MAX W W W Y Y Y Y Y Y + ? ? ? ? ? ? ? ?? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A , (1 125) AND ALSO FOR THE THERMAL MIXING LENGTH IN THE FOLLOWING FORM ( ) MAX MAX MAX 1 EXP PR T T W T T T Y Y Y B Y Y Y ? ? + ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? A , (1 126) WITH T ? = 0 43 TO 0 47 AND ( ) ( ) PR LOG PR I W I I B C ? = = ? , (1 127) WHERE, IN ACCORDANCE WITH NA AND HABIB (1973), THE C COEFFICIENTS ARE 34 96, 28 79, 33 95, 6 33 AND –1 186, RESPECTIVELY, REPRESENTING VERY GOOD DATA FOR HEAT TRANSFER IN A PIPE FOR RE > 10 000 AND PR = 0 02 TO 14 3 THIS IS EQUIVALENT TO ( ) 1 EXP PR 1 EXP PR T W Y Y B + + ? ? = ? ? ? ? ? ? (1 128) THIS IS MY FAVORITE APPROACH GRÖTZBACH (2007) PROPOSED TO USE INSTEAD THE RADIUS AS A NORMALIZING LENGTH SCALE THE DISTANCES MAX W Y , MAX TY FROM THE WALL AT WHICH THE VELOCITY AND THE TEMPERATURE HAVE THEIR MAXIMA, RESPECTIVELY 1 2 4 2 PRACTICAL RESULTS FROM THE ANALOGY BETWEEN MOMENTUM AND HEAT TRANSFER AN EXHAUSTIVE REVIEW OF THIS SUBJECT IS GIVEN BY KIRILLOV (1985, 1988) THE DATA COL- LECTION BY KIRILLOV CONSISTS OF MEASUREMENTS MADE BY DIFFERENT AUTHORS AND SHOWS A SPREAD OF ± 20% THE MAIN REASON FOR THE SPREAD IS THE SURFACE STRUCTURES, WHICH MAY DIFFER DUE TO DIFFERENT PHYSICO-CHEMICAL INFLUENCES THAT ARE DIFFICULT TO CONTROL THE SO- CALLED REYNOLDS ANALOGY PROVIDED THE USEFUL FRAMEWORK FOR CORRELATING THE DATA THE PIONEER ANALYTICAL SOLUTIONS IN THIS FIELD ARE GIVEN IN TABLE 1 3 USEFUL EMPIRICAL CORRE- LATIONS ARE SUMMARIZED IN TABLE 1 4 FROM THE 41 DISCUSSED CORRELATIONS KIRILLOV (1985, 1988) RECOMMEND THE SIMPLEST EXPRESSIONS VALID IN A NARROW PR-REGION 1 2 STEADY STATE FLOW IN PIPES WITH CIRCULAR CROSS SECTIONS TABLE 1 3 HEAT TRANSFER TO FLUID IN A PIPE DESCRIBED BY THE REYNOLDS ANALOGY RESULT VALID, METHOD, REFERENCE 48 NU = , W Q CONST ?? = , LAMINAR BAEHR AND STEPHAN (2004) P NU = , W T CONST = , LAMINAR BAEHR AND STEPHAN (2004) P REPR REPR 1 1 74RE PR FR NU ? ? = + ? PR < , TWO-LAYER MODEL, PRANDTL (1910) ( ) FR FR NU ? ? = ? ? ? ? + ?+ + ? ? ? ? ? ? ? ? ? PR < , SEMI-EMPIRICAL THREE- LAYERS MODEL, TAYLOR (1916), VON KARMAN (1939) ( ) ( ) ( ) 2/ REPR 8 PR FR FR FR FR NU F F ? ? ? ? = + ? ( ) FR FR F ? ? = + , ( ) 1/ 1 8 PR FR F ? = + , ( ) < < × , TWO-LAYER MODEL, PETUKHOV (1970), MEAN ERROR 1%, EXCEPT 0 0396RE PR 1 1 5RE PR PR ( ) 1/ < < × , MI- HEEV (1952) FOR VALI- DATION SEE FIG 3-23 IN MIHEEV AND MIHEEVA (1973) P 85 0 5EXP 0 6PR B = ? ? SLEICHER AND ROUSE (1975) 0 023RE PR PR PR 2 14RE PR < < , KU- TATELADZE (1979) ( ) FR FR NU ? ? ? ? ? = + ? ? ? ? ? ? ? , ( ) 1 82LOG RE FR ? ? = ? ? ? ? PETUKHOV AND KIRILLOV (1958) ( ) ( ) 1 1000/ RE FR FR NU ? ?