Mean flow structure of non-equilibrium boundary layers with adverse pressure gradient

Abstract

In this paper Spalding’s formulation for the law of the wall with constants modified by Persen is used to describe the inner region (viscous sub-layer and certain portion of logarithmic layer) and a wake law due to Persen is used to describe the wake region (outer region). These two laws are examined in the light of measured data by Marušić and Perry for non-equilibrium adverse pressure gradient layers. It is observed that structure of turbulence for this flow is well-described by these two laws. From the known structure of turbulence eddy viscosity for the flow under consideration is calculated. Self similarity in eddy viscosity is observed in the wall region.

Notations

A :

Constant appearing in Spalding’s formula

$A\left (x \right )$ :

Amplitude function

C _p :

Coefficient of pressure

P, p :

Pressure

Re _𝜃 :

Momentum thickness Reynolds number, U _o 𝜃/ν

r :

Pearson’s correlation coefficient

u :

Measured velocity

U _o :

Velocity at the edge of the boundary layer/free stream velocity

U _ref :

Reference velocity

u ⁺ :

Non-dimensional velocity, uv _∗/ν

$u_{\infty }^+$ :

Constant appearing in Persen’s wake law

$-\rho \overline {{u^{\prime }}{v}^{\prime }}$ :

Reynolds stress

x :

Down stream distance

y :

Distance from the wall

y ⁺ :

Non-dimensional wall distance, yv _∗/ν

$y_o^+$ :

Non-dimensional boundary layer thickness, δv _∗/ν

$u_1^+,y_1^+$ :

Coordinate of the meeting point of wall region and wake region

$w\left (\eta \right )$ :

Wake function

$w_{\max }$ :

Maximum value of wake

v _∗ :

Shear velocity

τ _t :

Turbulent shear stress

τ _w :

Shear stress at the wall, $\rho v_{\ast }^2$

ρ :

Density of fluid

κ :

Constant appearing in Spalding’s formulation

v :

Kinematic viscosity of fluid, μ/ρ

ν _e :

Kinematic eddy viscosity of fluid, μ _e /ρ

μ :

Viscosity of fluid

μ _e :

Eddy viscosity of fluid

ξ :

Non-dimensional velocity at the edge of the boundary layer, U _o /v _∗

δ :

Boundary layer thickness

δ ^∗ :

Displacement thickness, $\int _{0}^{\infty } {\left ({1-u/U_o } \right )dy}$

𝜃 :

Momentum thickness, $\int _{0}^{\infty } {\left ({u/U_o } \right )\left ({1-u/U_o } \right )} dy$

$\tan \gamma$ :

tangent on velocity profile

Λ :

Pressure parameter

β :

Clauser’s pressure parameter

π:

Coles’ wake parameter,

η :

Equals to y/δ = $y/\delta =y^+/y_o^+$.