Lecture by Dr. Bernhard Eisfeld

Dr.-Ing. Bernhard Eisfeld, DLR Braunschweig, Institut für Aerodynamik und Strömungstechnik, C²A²S²E Center for Computer Applications in AeroSpace Science and Engineering

Note: This presentation will form part as a lecture of the course Optimization in Fluid Mechanics. Everybody is welcome to attend though!

Mathematical Models of Viscous Fluid Flow


The presentation gives an introduction into the governing equations of viscous fluid flow, including turbulence and some basics of its modelling. After introducing Cartesian vectors and tensors and their index notation, the compressible Navier-Stokes equations will be derived, involving material laws and thermodynamic relations. Non-dimensionalization of the equations gives rise to similarity parameters, characterizing different types of flow.
For treating turbulent flows, simple and mass weighted averaging is applied, leading to the compressible Reynolds-averaged Navier-Stokes (RANS) equations. These equations contain new unknowns that need to be described by a turbulence model. The ideas of eddy-viscosity modelling will be presented, supplemented by a short characterization of standard models.


  • Cartesian Vectors and Tensors
  • Navier-Stokes Equations for Viscous Fluid Flow
    • Transport Equations
    • Material Laws
    • Thermodynamics
    • Wall Boundary Conditions
    • Non-Dimensional Navier-Stokes Equations
  • Averaged Flow Equations
    • Turbulent Flows
    • Averaging
    • RANS Equations
  • Statistical Turbulence Modelling
    • Boussinesq Hypothesis
    • Eddy-Viscosity Models
    • Turbulent Heat-Flux Modelling


  • R. Aris: „Vectors, Tenors, and the Basic Equations of Fluid Mechanics“, Dover, 1962.
  • F.M. White: „Viscous Fluid Flow“, McGraw-Hill, 1991.
  • G.K. Batchelor: „An Introduction to Fluid Dynamics“, Cambridge, 1967.
  • H. Schlichting: „Boundary Layer Theory“, 7th Edition, McGraw-Hill, 1979.
  • S.B. Pope: „Turbulent Flows“, Cambridge, 2000.
  • P.A. Durbin, B.A. Pettersson Reif: „Statistical Theory and Modeling for Turbulent Flows“, Wiley, 2001.
  • D.C. Wilcox: „Turbulence Modeling for CFD“, 3rd Edition, DCW Industries.
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