Direct Numerical Simulation of Equilibrium Turbulent Boundary Layers

  • Philippe R. Spalart
  • Anthony Leonard

Abstract

This paper describes the simulation of turbulent boundary layers by direct numerical solution of the three-dimensional time-dependent Navier-Stokes equations, using a spectral method. The flow is incompressible, and Re δ * = 1000 for most cases. The equations are written in the similarity coordinate system and normalized by the local friction velocity. Periodic streamwise and spanwise boundary conditions are then imposed. A family of nine “equilibrium” boundary layers, from the strongly accelerated “sink” flow to Stratford’s separating flow, is treated. Good general agreement with experiments is observed. The effects of the pressure gradient and of the Reynolds number are discussed.

Keywords

Vortex Anisotropy Convection Coherence Vorticity 

Nomenclature

a)

(XX 1) U X/U Bradshaw’s pressure gradient parameter

A)

Envelope function, see (6)

C)

Intercept, see (12 a)

D)

D u τ U x/U u τ x) Coles’ pressure gradient parameter

G)

(H ‒ 1) U (Hu τ ) Clauser’s shape factor

H)

δ*/θ shape factor

Reδ*)

δ* U /v Reynolds number, based on δ*

S)

y/∂x |n slope of the similarity lines

t)

Time

u, v, w)

Velocity components

U, V)

Mean velocity components

u’, v’, w’)

Fluctuating velocity components

up, vp, wp)

Periodic normahzed fluctuating components, see (6)

U)

Free-stream velocity

U)

Free-stream velocity

uτ)

Friction velocity

u+)

u/u τ wall units

x, y, z)

Stream wise, vertical, spanwise cartesian coordinates

X)

Distance along the boundary layer

X1)

Virtual origin in Bradshaw’s equilibrium boundary layer

y+)

yu τ /v wall units

α)

d τ /dy stress derivative at the wall in case 9

β)

U U */u 2 τ Clauser’s pressure gradient parameter

δ)

Characteristic thickness, see (1 b)

δ*)

Displacement thickness

ΔU+)

Wake component

η)

Vertical coordinate in similarity system

K)

Karman constant, see (12)

Δx, Δz)

Periods in x and z directions

v)

Kinematic viscosity

ω)

Vorticity

Ω)

Over-relaxation parameter

θ)

Momentum thickness

τ)

Shear stress

<.>)

Average taken over x, z and t

(τ, x, etc.)

Derivatives of ( with respect to t, x, etc.

2)

xx + yy + zz Laplace’s operator

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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Philippe R. Spalart
    • 1
  • Anthony Leonard
    • 1
  1. 1.NASA Ames Research CenterUSA

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