User:Egm6936.s11/CTIF

=Computational turbulent incompressible flow=

This work is a review of a reading of Computational Turbulent Incompressible Flow by Hoffman as part of the Body and Soul project. The book is a 'test' of the Generalized Galerkin (G2) method by applying it to the incompressible Navier-Stokes equations. Note that no turbulence modeling is performed, instead automatic stabalizing dissipation is introduced which acts as a turbulence model. This stabilization is addressed in Chapter 18, where the dissipation is shown to affect only the smallest unresolved scales.

Treatment of viscosity
The approach taken is to assume that the physical viscosity is less than the artificial viscosity so that the physical viscosity may remain unknown. Then if the solution converges as the artificial viscosity decreases a solution to the physical viscosity is found. The 'trick' of the method is to turn the disadvantage of artificial viscosity into an advantage for small physical viscosity.

Treatment of conduction
Conduction in turbulent flow is also assumed small and overcome by convective heat transfer. In this way the conduction coefficient may also remain unknown.

Two important assumptions so far in the text, small viscosity (Euler) and small conduction coefficients.

= Chapter 1 - introduction =

The introduction starts the first part of the book which serves the purpose to motivate the reader and introduce them to Fluids concepts. An undergraduate level understanding of fluids is more than sufficient for this text.

The chapter provides:


 * A description of a fluid and turbulence
 * Discussion on the history of Eueler and Navier-Stokes equations with regard to Leonhard Euler (1707-1783), Claude Louis Marie Henri Navier (1785-1836), George Gabriel Stokes (1819-1903), Siméon Denis Poisson (1781-1840), and Adhémar Jean Claude Barré de Saint-Venant (1797–1886)

Some discussion, mostly motivating, on the use of G2 for NS and Euler equations is made.
 * The main objective of solving the NS/Euler equations will be achieved by using the General Galerkin method. Actually, the author solves the Euler equation with numerical dissipation; the solution of the Euler equation with numerical dissipation would converge to the solution of the NS equation (with real, physical dissipation).
 * It should be noted the author takes a standpoint that turbulence is often sufficient to model features often assumed viscous features of NS (i.e. D'Alembart's phenomena)
 * A solution in finite element space with residual orthogonal to a set of finite element test functions is sought with a weighted least squares control of residual.
 * Automatic turbulence modeling
 * Automatic error control
 * Adaptivity is based on solving a linearized dual problem to obtain sensitivity with respect to the mesh.

= Chapter 2 - Mysteries and Secrets =

This chapter is motivating, with the intent of showing classical mysteries and how solving solutions of turbulent flow may resolve them.


 * Three paradoxes are discussed
 * d'Alembert's Mysteru: Zero drag of inviscid flow
 * Loschmidt's Mystery: Violation of the 2nd law of thermodynamics
 * Sommerfeld's Mysetery: Stability of Couette flow
 * Brief mention of the magnus effect

These effects are shown to be related to turbulence within the scope of Euler. Traditionally NS is used to justify these "viscous" effects.

= Chapter 3 - Turbulent Flow and History of Aviation =

This chapter provides a chronological development of aviation and turbulent flow. There is an interesting addition of Otto and Cayley that was unexpected, perhaps a European perspective.

= Chapter 4 - The Euler Equations =

This chapter is also introductory, but it more formally introduces the Euler equations. The Euler Equations are stated and derived in this chapter from a volume integral method.

In breif:

Consider a small fixed volume $$V$$ in $$\Omega$$ with the boundary $$S$$, then by mass conservation and the Divergence Theorem (covered in Body and Soul Vol 3) we derive for all Volumes, $$V$$.

$$ \begin{align} &\int_V\dot\rho dx = \frac{\partial }{\partial t}\int_V \rho dx = - \int_S (\rho u)\cdot n ds \\ &\int_V \nabla \cdot (\rho u) dx = \int_S ( \rho u)\cdot n ds \\ &\int_V \dot \rho dx + \int_V \nabla \cdot (\rho u)dx = 0 \end{align} $$

With continuous integrands we then have

$$\dot \rho + \nabla \cdot (\rho u) = 0 $$


 * The continuum assumption is discussed
 * The incompressibility is discussed and the Total Derivative (time plus spatial). Note that the derivation of the total derivative here was nice, with a chain rule perspective.

= Chapter 5 - The Incompressible Euler and Navier-Stokes Equations =

The Incompressible Euler Equations are discussed in this chapter along with the Navier-Stokes Equations. The issue of boundary conditions is discussed, in particular how the NS equations allow no slip walls, while Euler does not.


 * The Incompressible N-S Equations are discussed and the concept of a Newtonian Fluid is introduced. They note the modeling of the viscous term in the N-S equations and how this makes the N-S equations not ideal by Einstein's definition.  That is its not free of non-dimensional parameters (Re for example).
 * Viscosity is introduced and some words spent on justifying it being small, smaller than the artificial dissipation (specific to simulation of air?).
 * Friction (no slip) boundary conditions are discussed now that viscosity has been introduced.
 * Einstein's Ideal: self contained equations with no modeling or non-dimensional numbers.

ASIDE: They mention Einstein's biggest blunder here. But it should be noted that his biggest blunder, the introduction of a constant to make physical sense (he added a cosmological constant) turned out to be the way in which they are modelling dark matter/energy. Turns out his biggest blunder may be in assuming it a blunder.

It is also mentioned that these equations are just particular dynamic systems

= Chapter 6 - Triumph and Failure of Mathematics =

This chapter is also motivational, providing a historical perspective on the use of mathematics to model physical systems.

Laplace's formulation of Newton's theory of gravity in the form $$-\Delta \phi = \rho$$ was extremely successful and inspired the same approach in potential flow, $$\Delta \phi = 0$$, where the potential is the velocity potential and $$\nabla \phi$$ the velocity field (by irrotational flow).

This led to d'Alembert's drag paradox. Laplace's equation for fluid mechanics was a 'complete failure.'

A potential solution for fluids is not complete, nor as successful as it is for celestial mechanics.

= Chapter 7 - Laminar and Turbulent Flow =

This chapter discusses the transition from laminar to turbulent flow, and how the Reynolds number comes into play (and the issues with Reynolds number)

The onset of turbulent flow based on Reynolds number is elusive and varies from day-to-day. Chapter 36 will discuss why in detail. In short it's because "transition occurs if a product of perturbation growth and perturbation level is above a certain threshold, and only the perturbation growth can be connected to Re." In experiments, perturbation level varies from day-to-day. However, perturbations are always physically present, so turbulent flow is typical for Re>10^3.

For bluff bodies with high Reynolds numbers the wake region is turbulent. The boundary layer may be turbulent for very high (Re~10^6) Re the boundary layer may become turbulent and delay separation reducing the drag at higher speed (the drag crisis).

'''This book focuses on flows with medium Reynolds numbers (10^2 - 10^4) over large bodies (Re~10^-10^6) to very large (~10^7). They normalize to $$U=L=1$$, focusing on medium to small viscosity (v~10^-2-10^-4) to zero viscosity.'''

= Chapter 8 - Computational Turbulence =

Two interesting quotes extracted from the text:


 * I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of ﬂuids. And about the former I am rather optimistic. (Horace Lamb, 1932)


 * Consider a transport airplane with a 50-meter-long fuselage and wings with a chord length (the distance from the leading to the trailing edge) of about ﬁve meters. If the craft is cruising at 250 meters per second at an altitude of 10,000 meters, about 10 quadrillion (1016 ) grid points are required to simulate the turbulence near the surface with reasonable detail. (Parviz Moin and John Kim in Scientiﬁc American 1997)

In this chapter the difficulty of resolving turbulence is discussed (i.e., the need for modelling).

A rough DNS rule of thumb is $$Re^3$$ mesh points are needed in space-time, assuming the smallest scale in space and time is $$Re^{-3/4}$$. Thus, fully resolving a flow is not practical for many situations.

= Chapter 9 - A first study of stability =

In this chapter Euler solutions are shown to not be stable, and that the instability results in perturbations that evolve into turbulence, which itself is stabilizing. Thus the turbulent solution is stable.


 * Shows any Euler solution to be unstable by eigen analysis, and thus not practical.
 * Points out that Coutte flow is unstable because of the non-modal linear growth which leads to exponential growth. Sommerfeld assumed the non-modal growth was zero so that the flow was stable
 * Simulations of these exact solutions would evolve into turbulent flow.
 * Turbulence provides stability to the solution.

= Chapter 10 - d'Alembert's Mystery and Bernoulli's Law =

With stability and turbulent solutions introduce the mysteries first introduced are resolved.


 * Solution to d'Alembert's mystery is presented differently from Prandtl's (viscosity changes the flow in Prandtl's solution, whereas the author uses turbulence).
 * Some basic review of potential flow and Bernoulli's Law
 * Argues there is no stationary pointwise solution (turbulent solution instead).

= Chapter 11 - Prandtl's Resolution of d'Alembert's Mystery =

Prandtl's boundary layer discovery solved d'Alembert's Mystery. The author provides quotes from the original report and others on the topic to provide a traditional interpretation.

The author offers that stability with the vorticity generated at the boundary is the cause.

= Chapter 12 - New Resolution of d'Alembert's Mystery =

The author offers that the potential flow solution is not stable and instead a turbulent approximate solution develops with substantial drag. The drag results from the low pressure inside tubes of strong vorticity in the streamline direction. This is different than boundary layer effects of vanishing viscosity offered by Prandtl. The author leaves it to the reader to decide which explanation is more accurate.

G2 Simulations start appearing here where a simulation is shown of a sphere behaving physically without a boundary layer.

The author comments that the effect of the boundary layer should vanish with vanishing viscosity.


 * A stability analysis is done that shows vorticity is generated at the separation point. This part is worth reviewing.

The author sums with a turbulent solution to Euler resolves d'Alembert's Mystery without invoking boundary layer or NS.

= Chapter 13 - Turbulence and Chaos =

The author refers to the NS as a chaotic dynamical system (sensitive to perturbations). Some discussion on predictability is made with reference to weather prediction.

The challenge is to identify the order in the chaos (ie, longer term patterns)

The author argues that it is a matter of cancellation between all the possibilities (trajectories) that mean values become so predictable while shorter term values are unpredictable. It's said this is more than statistics.

A harmonic oscillatory is used as an example of a chaotic system. If the frequency is large relative to the averaging time period the value will be consistent. If the frequency is low, the average value will vary. This seems more a sampling issue than chaos theory, however.

A point is made that the time mean is comparable to a large ensemble of samplings. This appears to be leading to an average of turbulent flow in time is as good as taking many different measurements and finding a mean.

= Chapter 14 - A $1 Million Prize Problem =


 * There is a $1M prize for the existence, uniqueness and regularity of (pointwise) solutions to the NS equations for incompressible flow.
 * Jean Leray proved the existence of weak solutions, or turbulent solutions
 * By the nature of turbulent solutions the problem may not be solvable.


 * Well-posed: small variation in data results in small variations in the solution (Jacques Salomon Hadamard)"if very small changes in data could cause large changes in the solution, it would clearly be impossible to reach the basic requirement in science of reproducibility."
 * The author introduces a new concept of well-possed mean pointwise values "output uniqueness of approximate weak solutions"


 * The NS equations can be regularized many ways. Two ways are by introducing biLaplacian and a velocity dependent viscosity term.


 * The author shows weak uniqueness by computation after brief discussion on classical uniqueness efforts.

= Chapter 15 - Weak Uniqueness by Computation =


 * The author looks into convergence for flow around a cube with mixed results. Drag is found to be non-unique... I'm not sure I agree the specific test case having broader implications other than to show non-uniqueness.

= Chapter 16 - Existence of e-Weak solutions by G2 =


 * This chapter shows uniqueness in *stabalized* Galerkin finite element methods in the form of G2


 * An energy estimate is performed by multiplying the momentum equation by the velocity $$u$$, and integrating in space. Partial integration is used along with vector identities to bound the kinetic energy (conservation of energy).  Note, an integral in time remains to account for the dissipated energy through viscosity.  With a large dissipative term (about 1 for turbulent flow), a pointwise solution is expected.
 * G2 is a stabilized Galerkin method.
 * The existence of a solution is shown for the stabilized Galerkin method (bound on kinetic energy is used).
 * My concern is that a stabilized Galerkin solution (or stabilized NS for that matter) may not be a solution to the NS equations.

= Chapter 17 - Stability Aspects of Turbulence in Model Problems =


 * The author looks into stability properties of the linearized problem by looking into the real part of the eigenvalues, but notes that the imaginary contributions are not well understood at this point.
 * The author then looks at rotating flow results from G2 (and the eigenvalues).
 * The author also looks at flow over a cube.
 * Comments are made about ensemble values, in that the mean is relatively insensitive to the input variation/distribution.

= Chapter 18 - A convection-diffusion Model Problem =

Important: Here the issue of least squares stabilization of G2 introducing an artificial viscosity is addressed. The artificial viscosity acts as a turbulent diffusion on the smallest scales only, and therefore does not degrade the solution.

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