Compressible Frictional Flow Past Wings P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A Small and Significant Region of Curse ……..

A continuously Growing Solid affected Region. The Boundary Layer An explicit Negligence by Potential Flow Theory. Great Disadvantage for Simple fluid Systems

De Alembert to Prandtl Ideal to Real

Concept of Solid Fluid Interaction Diffuse reflection U 2 U U Φ U2 U2 Φ U1 U1 U1U1 Φ U2U2 Specular reflection Perfectly smooth surface (ideal surface) Real surface The Momentum & convective heat transfer is defined for a combined solid and fluid system. The fluid packets close to a solid wall attain a zero relative velocity close to the solid wall : Momentum Boundary Layer.

The fluid packets close to a solid wall come to mechanical equilibrium with the wall. The fluid particles will exchange maximum possible momentum flux with the solid wall. A Zero velocity difference exists between wall and fluid packets at the wall. A small layer of fluid particles close the the wall come to Mechanical, Thermal and Chemical Equilibrium With solid wall. Fundamentally this fluid layer is in Thermodynamic Equilibrium with the solid wall.

Introduction A boundary layer is a thin region in the fluid adjacent to a surface where velocity, temperature and/or concentration gradients normal to the surface are significant. Typically, the flow is predominantly in one direction. As the fluid moves over a surface, a velocity gradient is present in a region known as the velocity boundary layer, δ(x). Likewise, a temperature gradient forms (T ∞ ≠ T s ) in the thermal boundary layer, δ t (x), Therefore, examine the boundary layer at the surface (y = 0). Flat Plate Boundary Layer is an hypothetical standard for initiation of basic analysis.

Boundary Layer Thickness

The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:

Boundary Conditions 0 0 T wall VeVe 0 TT

Define dimensionless variables: Similarity Parameters:

Similarity Solution for Flat Plate Boundary Layer Similarity variables :

Substitute similarity variables: This is called as Blasius Equation. An ordinary differential equation with following boundary conditions.

Numerical Solution of Blasius Equation

Substitute in Blasius Equation

Fourth-order Runge-Kutta method

Blasius Solution  ff’f’’

Blasius Similarity Solution Blasius equation was first solved numerically (undoubtedly by hand 1908). Conclusions from the Blasius solution:

Variation of Reynolds numbers All Engineering Applications

What Sort of Reynolds Numbers do We Encounter in Supersonic Flight? “Transition Line” Space Shuttle

Velocity Profile in Boundary Layer

Simple Velocity Profile Models Laminar Turbulent

Turbulent Skin Friction Turbulent Boundary Layer No Theoretical Prediction for Boundary Layer Thickness for Turbulent Boundary layer Statistical Empirical Correlation “Time averaged”

Compare Laminar and Turbulent Skin Friction Plot these Formulae Versus R e

Compare Laminar and Turbulent Skin Friction

Comparison of Reynolds Number and Mach Number Reynold’s number is a measure of the ratio of the Inertial Forces acting on the fluid -- to -- the Viscous Forces Acting on the fluid -- Fundamental Parameter of Viscous Flow Mach number is a measure of the ratio of the fluid Kinetic energy to the fluid internal energy (direct motion To random thermal motion of gas molecules) -- Fundamental Parameter of Compressible Flow 2   V e 2 2   V e R e  c  V e   c  V e 2  V e  c  2 c   V e 2  w  Inertial  Forces Viscous  Forces

Empirical Skin Friction Correlations R e ~500,000 M=0 “Smooth Flat Plate with No Pressure Gradient”

Empirical Skin Friction Correlations “Smooth Flat Plate with No Pressure Gradient”

Plot the laws “exact solution for Laminar Flo”

Simple High Speed Skin Friction Model So For Our Purposes … we’ll use the “1/7 th power” Boundary layer law … and the Exact Laminar Solution Exact Blasius Solution R e < 500,000

Comparison of Velocity Distributions

Supersonic Boundary Layers When a vehicle travels at Mach numbers greater than one, a significant temperature gradient develops across the boundary layer due to the high levels of viscous dissipation near the wall. In fact, the static-temperature variation can be very large even in an adiabatic flow, resulting in a low density, high-viscosity region near the wall. In turn, this leads to a skewed mass-flux profile, a thicker boundary layer, and a region in which viscous effects are somewhat more important than at an equivalent Reynolds number in subsonic flow. Intuitively, one would expect to see significant dynamical differences between subsonic and supersonic boundary layers. However, many of these differences can be explained by simply accounting for the fluid-property variations that accompany the temperature variation, as would be the case in a heated incompressible boundary layer. This suggests a rather passive role for the density differences in these flows, most clearly expressed by Morkovin’s hypothesis.

Effect of Mach Number The friction coefficient is affected by Mach number as well. This effect is small at subsonic speeds, but becomes appreciable for supersonic aircraft. The idea is that aerodynamic heating modifies the fluid properties. For a fully-turbulent flow, the wall temperature may be estimated from: An effective incompressible temperature ratio is defined:

GAS Viscosity Models Sutherland’s Formula Result from kinetic theory that expresses viscosity as a function of temperature

C D fric     compressible    T  T avg cV   ( T  ) T avg T        3/2 T   C s T avg  C s                     1 7    cV   ( T  ) T  T avg       5/2 T avg  C s T   C s               1 7     cV   ( T  )       1 7 T  T avg       5/2 T avg  C s T   C s             1 7  C D fric     compressible  C D fric     incompressible T  T avg       5/2 T avg  C s T   C s               1 7

What is “T avg ” in the boundary layer?

Look at small segment of boundary layer, dy Enthalpy Balance

Taking the average (Integrating Across Boundary layer) yT avg  T   1  V  2 2 c p 1  u ( y ) V e       2       d 0    T   V  2 2 c p 1   2 7       dy 0 1   1    T   V  2 2 c p 1    /7           T   2 9 V  2 2 c p  T  1  2 9  1 2 M  2             Valid for Turbulent Flow

Collected Algorithm Valid for Turbulent Flow

Skin Friction Versus Mach Number

Symmetric Double-wedge Airfoil … L/D (revisited) + t/c = Inviscid Analysis Mach 3 t/c

Symmetric Double-wedgeAirfoil … L/D (revisited) = Analysis Including skin Friction Model Mach 3 L/D max =7.4 Blow up of Previous page

= Mach km Altitude L/D max =3.18