# Boundary-Layer Theory

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1 Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22 Tables Springer

2 Introduction XIX Part I. Fundamentals of Viscous Flows 1. Some Features of Viscous Flows Real and Ideal Fluids Viscosity Reynolds Number Laminar and Turbulent Flows Asymptotic Behaviour at Large Reynolds Numbers Comparison of Measurements Using the Inviscid Limiting Solution Summary Fundamentals of Boundary Layer Theory Boundary-Layer Concept Laminar Boundary Layer on a Flat Plate at Zero Incidence Turbulent Boundary Layer on a Flat Plate at Zero Incidence Fully Developed Turbulent Flow in a Pipe Boundary Layer on an Airfoil Separation of the Boundary Layer Overview of the Following Material Field Equations for Flows of Newtonian Fluids Description of Flow Fields Continuity Equation Momentum Equation General Stress State of Deformable Bodies General State of Deformation of Flowing Fluids Relation Between Stresses and Rate of Deformation Stokes Hypothesis Bulk Viscosity and Thermodynamic Pressure Navier-Stokes Equations 68

3 X 3.10 Energy Equation Equations of Motion for Arbitrary Coordinate Systems (Summary) Equations of Motion for Cartesian Coordinates in Index Notation Equations of Motion in Different Coordinate Systems General Properties of the Equations of Motion Similarity Laws Similarity Laws for Flow with Buoyancy Forces (Mixed Forced and Natural Convection) Similarity Laws for Natural Convection Vorticity Transport Equation Limit of Very Small Reynolds Numbers Limit of Very Large Reynolds Numbers Mathematical Example of the Limit Re > oo Non-Uniqueness of Solutions of the Navier-Stokes Equations Exact Solutions of the Navier-Stokes Equations Steady Plane Flows Couette-Poiseuille Flows Jeffery- Hamel Flows (Fully Developed Nozzle and Diffuser Flows) Plane Stagnation-Point Flow Flow Past a Parabolic Body Flow Past a Circular Cylinder Steady Axisymmetric Flows Circular Pipe Flow (Hagen-Poiseuille Flow) Flow Between Two Concentric Rotating Cylinders Axisymmetric Stagnation Point Flow Flow at a Rotating Disk Axisymmetric Free Jet Unsteady Plane Flows Flow at a Wall Suddenly Set into Motion (First Stokes Problem) Flow at an Oscillating Wall (Second Stokes Problem) Start-up of Couette Flow Unsteady Asymptotic Suction Unsteady Plane Stagnation-Point Flow Oscillating Channel Flow Unsteady Axisymmetric Flows Vortex Decay Unsteady Pipe Flow Summary 141

4 XI Part II. Laminar Boundary Layers 6. Boundary-Layer Equations in Plane Flow; Plate Boundary Layer Setting up the Boundary-Layer Equations Wall Friction, Separation and Displacement Dimensional Representation of the Boundary-Layer Equations Friction Drag Plate Boundary Layer General Properties and Exact Solutions of the Boundary-Layer Equations for Plane Flows Compatibility Condition at the Wall Similar Solutions of the Boundary-Layer Equations Derivation of the Ordinary Differential Equation 167 A Boundary Layers with Outer Flow 169 B Boundary Layers Without Outer Flow Wedge Flows Flow in a Convergent Channel Mixing Layer Moving Plate Free Jet Wall Jet Coordinate Transformation Gortler Transformation v. Mises Transformation Crocco Transformation Series Expansion of the Solutions Blasius Series Gortler Series Asymptotic Behaviour of Solutions Downstream Wake Behind Bodies Boundary Layer at a Moving Wall Integral Relations of the Boundary Layer Momentum Integral Equation Energy-Integral Equation Moment-of-Momentum Integral Equations Approximate Methods for Solving the Boundary Layer Equations for Steady Plane Flows Integral Methods 196

5 XII 8.2 Stratford's Separation Criterion Comparison of the Approximate Solutions with Exact Solutions Retarded Stagnation-Point Flow Divergent Channel (Diffuser) Circular Cylinder Flow Symmetric Flow past a Joukowsky Airfoil Thermal Boundary Layers Without Coupling of the Velocity Field to the Temperature Field Boundary-Layer Equations for the Temperature Field Forced Convection for Constant Properties Effect of the Prandtl Number Similar Solutions of the Thermal Boundary Layer Integral Methods for Computing the Heat Transfer Effect of Dissipation; Distribution of the Adiabatic Wall Temperature Thermal Boundary Layers with Coupling of the Velocity Field to the Temperature Field Remark Boundary- Layer Equations Boundary Layers with Moderate Wall Heat Transfer (Without Gravitational Effects) Perturbation Calculation Property Ratio Method (Temperature Ratio Method) Reference Temperature Method Compressible Boundary Layers (Without Gravitational Effects) Physical Property Relations Simple Solutions of the Energy Equation Transformations of the Boundary-Layer Equations Similar Solutions Integral Methods Boundary Layers in Hypersonic Flows Natural Convection Boundary Layer Equations Transformation of the Boundary-Layer Equations Limit of Large Prandtl Numbers (T w = const) Similar Solutions General Solutions Variable Physical Properties Effect of Dissipation 280

6 XIII 10.6 Indirect Natural Convection Mixed Convection Boundary-Layer Control (Suction/Blowing) Different Kinds of Boundary-Layer Control Continuous Suction and Blowing Fundamentals Massive Suction Massive Blowing Similar Solutions General Solutions Plate Flow with Uniform Suction or Blowing Airfoil Natural Convection with Blowing and Suction Binary Boundary Layers Overview Basic Equations Analogy Between Heat and Mass Transfer 316 ' Similar Solutions Axisymmetric and Three-Dimensional Boundary Layers Axisymmetric Boundary Layers Boundary Layer Equations Mangier Transformation Boundary Layers on Non-Rotating Bodies of Revolution Boundary Layers on Rotating Bodies of Revolution Free Jets and Wakes Three-Dimensional Boundary Layers Boundary-Layer Equations Boundary Layer at a Cylinder Boundary Layer at a Yawing Cylinder Three Dimensional Stagnation Point Boundary Layers in Symmetry Planes General Configurations Unsteady Boundary Layers Fundamentals Remark Boundary-Layer Equations Similar and Semi-Similar Solutions Solutions for Small Times (High Frequencies) Separation of Unsteady Boundary Layers Integral Relations and Integral Methods Unsteady Motion of Bodies in a Fluid at Rest 355

7 XIV Start-Up Processes Oscillation of Bodies in a Fluid at Rest Unsteady Boundary Layers in a Steady Basic Flow Periodic Outer Flow Steady Flow with a Weak Periodic Perturbation Transition Between Two Slightly Different Steady Boundary Layers Compressible Unsteady Boundary Layers Remark Boundary Layer Behind a Moving Normal Shock Wave Flat Plate at Zero Incidence with Variable Free Stream Velocity and Wall Temperature Extensions to the Prandtl Boundary-Layer Theory Remark Higher Order Boundary-Layer Theory Hypersonic Interaction Triple-Deck Theory Marginal Separation Massive Separation 408 Part III. Laminar Turbulent Transition 15. Onset of Turbulence (Stability Theory) Some Experimental Results on the Laminar-Turbulent Transition Transition in the Pipe Flow Transition in the Boundary Layer Fundamentals of Stability Theory Remark Fundamentals of Primary Stability Theory Orr Sommerfeld Equation Curve of Neutral Stability and the Indifference Reynolds Number 434 a Plate Boundary Layer 436 b Effect of Pressure Gradient 445 c Effect of Suction 457 d Effect of Wall Heat Transfer 460 e Effect of Compressibility 463 f Effect of Wall Roughness 467 g Further Effects Instability of the Boundary Layer for Three-Dimensional Perturbations 473

8 Remark Fundamentals of Secondary Stability Theory Boundary Layers at Curved Walls Boundary Layer at a Rotating Disk Three-Dimensional Boundary Layers Local Perturbations 491 XV Part IV. Turbulent Boundary Layers 16. Fundamentals of Turbulent Flows Remark Mean Motion and Fluctuations Basic Equations for the Mean Motion of Turbulent Flows Continuity Equation Momentum Equations (Reynolds Equations) ,3 Equation for the Kinetic Energy of the Turbulent Fluctuations (/c-equation) Thermal Energy Equation Closure Problem Description of the Turbulent Fluctuations Correlations Spectra and Eddies Turbulence of the Outer Flow Edges of Turbulent Regions and Intermittence Boundary Layer Equations for Plane Flows Internal Flows Couette Flow Two-Layer Structure of the Velocity Field and the Logarithmic Overlap Law Universal Laws of the Wall Friction Law Turbulence Models Heat Transfer Fully Developed Internal Flows (A const) Channel Flow Couette-Poiseuille Flows Pipe Flow Slender-Channel Theory Turbulent Boundary Layers Without Coupling of the Velocity Field to the Temperature Field Turbulence Models 557

9 XVI Remark Algebraic Turbulence Models Turbulent Energy Equation Two-Equation Models Reynolds Stress Models Heat Transfer Models Low-Reynolds-Number Models Large-Eddy Simulation and Direct Numerical Simulation Attached Boundary Layers Layered Structure Boundary-Layer Equations Using the Defect Formulation Friction Law and Characterisitic Quantities of the Boundary Layer Equilibrium Boundary Layers Boundary Layer on a Plate at Zero Incidence Boundary Layers with Separation Stratford Flow Quasi-Equilibrium Boundary Layers Computation of Boundary Layers Using Integral Methods Direct Method Inverse Method Computation of Boundary Layers Using Field Methods Attached Boundary Layers Boundary Layers with Separation Low-Reynolds-Number Turbulence Models Additional Effects Computation of Thermal Boundary Layers Fundamentals Computation of Thermal Boundary Layers Using Field Methods Turbulent Boundary Layers with Coupling of the Velocity Field to the Temperature Field Fundamental Equations Time Averaging for Variable Density Boundary-Layer Equations Compressible Turbulent Boundary Layers Temperature Field Overlap Law Skin-Friction Coefficient and Nusselt Number Integral Methods for Adiabatic Walls 623

10 XVII Field Methods Shock-Boundary-Layer Interaction Natural Convection Axisymmetric and Three-Dimensional Turbulent Boundary Layers Axisymmetric Boundary Layers Boundary-Layer Equations Boundary Layers Without Body Rotation Boundary Layers with Body Rotation Three-Dimensional Boundary Layers Boundary-Layer Equations Computation Methods Examples Unsteady Turbulent Boundary Layers Averaging and Boundary-Layer Equations Computation Methods Examples Turbulent Free Shear Flows Remark Equations for Plane Free Shear Layers Plane Free Jet Global Balances Far Field Near Field Wall Effects Mixing Layer Plane Wake Axisymmetric Free Shear Flows Basic Equations Free Jet Wake Buoyant Jets Plane Buoyant Jet Axisymmetric Buoyant Jet Plane Wall Jet 677

11 XVIII Part V. Numerical Methods in Boundary-Layer Theory 23. Numerical Integration of the Boundary Layer Equations Laminar Boundary Layers Remark Note on Boundary-Layer Transformations Explicit and Implicit Discretisation Solution of the Implicit Difference Equations Integration of the Continuity Equation Boundary-Layer Edge and Wall Shear Stress Integration of the Transformed Boundary-Layer Equations Using the Box Scheme Turbulent Boundary Layers Method of Wall Functions Low-Reynolds-Number Turbulence Models Unsteady Boundary Layers 701 Z 23.4 Steady Three-Dimensional Boundary Layers 703 List of Frequently Used Symbols 709 References 717 Index 795

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