Invariant 2-equation modeling

A Consistent 4D Invariant Turbulence Modeling Approach

Michael Frewer Technical University of Darmstadt Chair of Fluid Dynamics Department of Mechanical Engineering

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 1/25

The aim of this talk (1) To demonstrate that  within Newtonian physics modeling turbulence on a 3D spatial geometry is not equivalent to modeling turbulence on a true 4D space-time geometry  Consequence: to model non-stationary effects  Methodology: Differential geometry

 next to velocity gradients also pressure gradients can be consistently introduced and used as closure variables during any modeling process  Consequence: to model non-local effects  Methodology: Lie-group symmetry analysis 7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 2/25

The aim of this talk (2) To apply both ideas  in the most simplest modeling environment  1-point statistics using the RANS concept

 to construct a qualitatively new 4D nonlinear EVM within the family for high turbulent Reynolds numbers  considering only the lowest (quadratic) nonlinearity [M. Frewer (2009), JFM]

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 3/25

4D vs. (3+1)D Modeling (1) In a 4D space-time setting  variables of space and time are fully independent  in any closure strategy not only space but also time derivatives have to be considered, hence allowing for a universal and consistent treatment of curvature and non-stationary effects

 physical quantities as velocities, forces or stresses transform as tensors, irrespective of whether they are objective, i.e. frame-independent, or not  this is important in particular when to model unclosed terms with non-objective quantities

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 4/25

4D vs. (3+1)D Modeling (2)  frame accelerations or inertial forces of any kind within Newtonian physics can always be interpreted as a pure geometrical effect  a true 4D model will describe non-inertial turbulence equally well or equally bad as the corresponding inertial case

 the special space-time structure of the 4D manifold allows for additional modeling constraints, which are absent in the usual (3+1)D geometrical formulation  e.g. within the 4D manifold averaged and fluctuating velocities evolve differently: as time-like vectors, as space-like vectors

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 5/25

Universal Form-Invariance (UFI) One needs a new mathematical framework which allows for universal form invariance under time-dependent coordinate transformations

Def.: Form-invariance (Covariance): Structural invariance Frame-independence (Objectivity): Symmetric invariance Note: Frame-independence

Form-invariance

Inertial frames: Frame-independence

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 6/25

Form-invariance

Minimal requirements for UFI in Newtonian Physics Aim is to construct a true 4D space-time manifold  with the most simplest geometrical structure  in which physics can evolve on the basis of a Newtonian description  and in which it is possible to measure distances in time and space, i.e. should be endowed with a metric

Is that possible ?

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 7/25

Historical Note on UFI (1) First universal form-invariant theory: Einstein‘s Theory of General Relativity (1916) as a Relativistic Theory of Gravitation built on three physical axioms: I. General covariance (UFI) II. Constant speed of light in all local inertial reference frames III. Equivalence between inertial and gravitational mass

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 8/25

Historical Note on UFI (2)  Kretschmann‘s Objection (1917): Axiom I. is physically vacuous Any theory can be put in generally covariant form  Trivial ojection for already existing physical theories  Highly non-trivial objection when constructing new physical theories, like GRT

 Cartan & Friedrich (1925): Newton-Cartan Theory of Gravitation Only Axiom II. is a characteristic feature of Einsteinian mechanics  Newtonian mechanics can always be reformulated such that Axioms I. & III. are satified, but then  Physics has to evolve in a non-Riemannian space-time manifold

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 9/25

Construction of a 4D Newtonian spacetime manifold (1)  Ideas of Cartan & Friedrich were never used beyond the theory of gravitation  Task: To generally prepare it for classical continuum mechanics in the limit of small mass scales, i.e. gravitation is then decoupled from the space-time geometry [M. Frewer (2008), Acta Mechanica]

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 10/25

Construction of a 4D Newtonian spacetime manifold (2) Thus the minimal requirements for chooses

can be fulfilled if one

 A manifold with zero curvature  i.e. there exists a global coordinate system in which the affine connection vanishes

 Newtonian mechanics to emerge from Einsteinian mechanics in the classical limit  the limit is to be taken on the Minkowskian manifold of special relativity with the pseudo-Euclidean metric

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 11/25

Structure of the 4D Newtonian spacetime manifold (1) Results: The 4D Newtonian space-time manifold  is a non-Riemannian manifold with a non-unique and singular metrical connection  four singular metrics which can only connect pure space-like or pure time-like quantities, but no variants thereof

(Representation: Cartesian coordinates) 7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 12/25

Structure of the 4D Newtonian spacetime manifold (2)  only allows for space-time coordinate transformations in which the time coordinate transforms as an absolute quantity  Euclidean transformations only form a small subset

 the 4-velocity has the structure  velocity field is always a time-like quantity

and then define:

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 13/25

Procedure for UFI Since our aim is to achieve UFI for already existing physical theories in a flat manifold, the procedure is simple and defined as follows: 1. Write the Newtonian equations in the inertial (3+1)D Cartesian form 2. Rewrite them into the corresponding 4D form using the geometrical structure of the Newtonian space-time manifold 3. Make the transition from inertial Cartesian to arbitrary spacetime coordinates by replacing 7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 14/25

UFI of the Navier-Stokes equations (1)

The physical content of the theory is not changed! 7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 15/25

UFI of the Navier-Stokes equations (2)

 By construction: These equations stay form-invariant under arbitrary space-time transformations with

 The action of the covariant derivative:

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 16/25

UFI of the ensemble-averaged NavierStokes equations

 The average 4-velocity vector,  while the fluctuating 4-velocity vector

is a pure time-like

is a pure space-like

 Thus the Reynolds-stress tensor is a pure spacelike quantity, which has to be respected during modelling 7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 17/25

Ansatz for Invariant Modeling (1) The aim is to close the Reynolds-stress tensor algebraically

The modeling restrictions for

are:

1) contravariant tensor of rank 2 2) pure space-like tensor 3) symmetric tensor 4) carries the space-like dimension

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 18/25

Ansatz for Invariant Modeling (2) Heuristics:  The most basic Ansatz for the closure set: where

 A prior Lie-group symmetry analysis excludes from :

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 19/25

and

Invariant 2-equation modeling (1)  Next to the Reynolds-stress tensor 6 additional unclosed quantities have to be modelled, resulting from the transport equation  for the invariant turbulent kinetic energy

 and its invariant (pseudo-)dissipation rate

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 20/25

Invariant 2-equation modeling (2)  … where

 …

 …  …

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 21/25

is time-like for

Proposal for an invariant high-Re turbulence model (1)

 Lie-group symmetry analysis (without ICs & BCs):

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 22/25

Proposal for an invariant high-Re turbulence model (2)

 Tensor Invariant Theory: (Spencer & Rivlin 1958)  Simplification: For an overall quadratic nonlinearity, the expansion  in the two production terms has to be truncated at linear order  for the remaining 4 unclosed tensors at quadratic order

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 23/25

Results (1)  The appearance of two time-like invariants (to model memory effects):

 Time-like terms in the expansion for the space-like diffusion vectors

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 24/25

Results (2)  A quadratic expansion in the Reynolds-stress tensor already allows for a term being able to capture secondary flow effects

 Example: axially rotating pipe flow with rotation rate

 In current nonlinear eddy viscosity models a higher non-linearity is needed  In general: The mean pressure gradient shows itself as a promising closure variable to model non-local effects

7 August 2009 | CTR Seminar | Stanford University | M. Frewer | 25/25