Direct Numerical Simulation of Equilibrium Turbulent Boundary Layers

  • Philippe R. Spalart
  • Anthony Leonard


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.


Boundary Layer Reynolds Number Direct Numerical Simulation Reynolds Stress Turbulent Boundary Layer 
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(XX 1) U X/U Bradshaw’s pressure gradient parameter


Envelope function, see (6)


Intercept, see (12 a)


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


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


δ*/θ shape factor


δ* U /v Reynolds number, based on δ*


y/∂x |n slope of the similarity lines



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)


Free-stream velocity


Free-stream velocity


Friction velocity


u/u τ wall units

x, y, z)

Stream wise, vertical, spanwise cartesian coordinates


Distance along the boundary layer


Virtual origin in Bradshaw’s equilibrium boundary layer


yu τ /v wall units


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


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


Characteristic thickness, see (1 b)


Displacement thickness


Wake component


Vertical coordinate in similarity system


Karman constant, see (12)

Δx, Δz)

Periods in x and z directions


Kinematic viscosity




Over-relaxation parameter


Momentum thickness


Shear stress


Average taken over x, z and t

(τ, x, etc.)

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


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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