NASA Technical Memorandum 102847 _
A One-EquationTurbulenceTransport
Model for High Reynolds Number
Wall-Bounded Flows
Barrett S. Baldwin and Timothy J. Barth
August 1990
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NASA Technical Memorandum 102847
A One-EquationTurbulenceTransport
Model for High ReynoldsNumber
Wall-BoundedFlows
Barrett S. Baldwin and Timothy J. Barth, Ames Research Center, Moffett Field, California
August 1990
National Aeronautics and
Space AcIministratbn
Ames Research Center
Moffett Field, California 94035-1000
SUMMARY
A one-equation turbulence model that avoids the need for an algebraic length scale
is derived from a simplified form of the standard k - c model equations. After calibration
based on well established properties of the flow over a flat plate, predictions of several
other flows are compared with experiment. The preliminary results presented indicate that
the model has predictive and numerical properties of sufficient interest to merit further
investigation and refinement. The one-equation model is also analyzed numerically and
robust solution methods are presented.
INTRODUCTION
One motivation for the developments documented in this report was the inability
of well-established Navier-Stokes solvers using algebraic turbulence models to adequately
predict several of the turbulent flow fields contained in the Viscous Transonic Airfoil Work-
shopt. These flows contained significant separated flow and uniformly poor results were
reported by all participants using the algebraic turbulence models of Baldwin-Lomax (ref.
1) or Cebeci-Smith (ref. 2). (The results predicted by the ARC2D code were reported
by Maksymiuk and PuUiam in ref. 3.) Two cases were adequately predicted by only one
turbulence model, that of Johnson and King (ref. 4) as reported by King (ref. 5) and
verified by Coakley (ref. 6) using an independent code.
Another motivation is the need to treat flow problems in which multiple shear layers
are present such that the determination of algebraic length scales is cumbersome and un-
reliable. An example is the Coanda airfoil configuration computed by Pulliam, Jespersen,
and Barth (ref. 7) which exploits tangential surface blowing. The need to avoid algebraic
length scales leads to a consideration of k - _ or related two-equation models.
From our limited experience with two-equation models and reports by others (e.g.,
Sugavanum (ref. 8)) it became apparent that it would be worthwhile to investigate the
possibility of transforming to variables other than the basic physical variables in an effort
to avoid the well-known numerical difficulties that occur in the solution of the standard
k - _ equations. In the course of that investigation, a self-consistent one-equation model
was found that also avoids the need for algebraic length scales. The main purpose of this
report is to present the one-equation model and show the applicability of the model to a
range of difficult turbulent flows. Results from computations of the two troublesome cases
in the Viscous Transonic Airfoil Workshop are reported and significant improvement is
achieved.
The following two sections explain the rationale behind the development of the one-
equation model. The next contains the solution for a self-similar turbulent wake to demon-
strate a degree of generality of the model. In the overall design of the model equation, em-
phasis was placed on numerical considerations so that extremely robust numerical solution
methods could be used. The Numerical Implementation section gives some of the numer-
ical theory and analysis needed to properly incorporate the one-equation model into flow
solvers. The one-equation model has been implemented in a number of central-difference
t Held in conjunction with the AIAA 25th Aerospace Sciences Meeting (January 1987).