ME 634-001
FINAL EXAMINATION
Spring 2015
(300 pts.)
Closed book(s), closed notes, no electronic devices of any type permitted; and no barbecuing,
drinking or fighting. Good Luck!
Part I. Definitions, short-answer questions, etc. (10 pts. each)
1. Sketch a hypothetical stationary turbulent time series. Then sketch corresponding lowpass, high-pass and time-averaged time series.
2. List the three (3) main results of Kolmogorov’s theory, and very briefly note the importance
of each.
3. Briefly compare the three main approaches currently in use for turbulence calculations. In
particular, state the degree of modeling required, and give (do not derive) an estimate of
total required machine arithmetic for each.
4. The N.–S. equations are now known to exhibit at least three short bifurcation sequences
leading to chaotic, turbulent behaviors. State these, and give names of researchers who
were responsible for first identifying them.
5. Sketch the 1-D energy spectrum (as a function of wavenumber) for incompressible N.–S.
turbulence. Mark this to indicate the main length scales and physical regions of importance
in analysis of turbulent flows.
6. Describe the relationship, if any, between randomness and stochasticity. What does one
observe in autocorrelations corresponding to these behaviors?
7. Write the formula that defines a scalar structure function of order p, and give the corresponding Kolmogorov power law when p = 2.
8. Discuss validity, or lack thereof, of the oft-quoted assertion, “Eddy viscosity explains
enhanced momentum transport in turbulent flows.”
9. Contrast the notions of implicit and explicit filtering in the context of LES. Note what is
filtered in each case and the consequences of this.
10. State the mathematical, but non-rigorous, definition of a strange attractor. Provide qualitative descriptions of each part of this definition.
Part II. Longer problems, derivations, etc. (point value in parentheses, 200 total pts.)
(30) 1. Derive the log law in terms of u+ and y+ , and specifically give their definitions in the
course of the derivation. Then plot these with qualitatively correct axes. Annotate this
to show regions corresponding to the viscous sublayer, buffer layer, log layer and wake (or
defect) region.
1
(25) 2. Construct a Fourier–Galerkin approximation to the 1-D Burgers’ equation,
ut + u2 x = −px + νuxx , (x, t) ∈ (0, 1)×(t0 , tf ] ,
with initial data
u(x, t0 ) = u0 (x) ,
and boundary conditions
u(0, t) = u(1, t) = 0 ∀ t ,
using an arbitrary (do not select a particular one) complex exponential-like basis set
{ϕk (x)}∞
k=k0 chosen to satisfy the Dirichlet boundary conditions, be orthonormal and be
complete in L2 (0, 1). Assume px (x, t) is a given function in L2 (0, 1) which is continuous
with respect to time, t; the constant ν > 0 is also given.
(a) Write the Fourier representations for u(x, t) and px (x, t). Also give formulas for
calculating the Fourier coefficients of px and u0 (x).
(b) Derive the Galerkin ODEs for the Fourier coefficients of u(x, t).
(c) Identify the dissipative part of these equations, and note the physics represented by
analogous terms in the N.–S. equations. Discard all nonlinear and forcing terms from
the Galerkin ODEs, and solve the resulting equation(s) for a single Fourier coefficient.
Discuss the behavior of this solution as k → ∞ for fixed ν and as ν → 0 for fixed
wavenumber k, particularly describing the mathematical/physical consequences.
(d) Solve the ODE corresponding to a single quadratic mode from the set of nonlinear
terms (again, in the absence of forcing). Justify the simplifications you would employ
to obtain this equation in the N.–S. context, and discuss implications of its solution.
(15) 3. Derive Kolmogorov length, time and velocity scales, and give the value (a number) of the
Kolmogorov-scale Reynolds number.
(20) 4. Compare and contrast Reynolds and LES decompositions of a velocity component, say
u(x, t), by carrying out the following steps.
(a) Write the Reynolds decomposition of u(x, t), and give formulas defining each of its
terms, assuming u(x, t) is known. Prove the well-known basic properties of these
terms with respect to time averaging.
(b) Repeat (a) for LES decomposition of u(x, t) with time averaging replaced by spatial
filtering using an arbitrary, unspecified (but not projective) filter.
(c) Expand the decompositions of parts (a) and (b) in Fourier series, and discuss the
differences with respect to construction of models for the fluctuating parts.
(20) 5. Provide sketches and discussions of the geometric construction of a strange attractor. In
particular, note how these are related to the non-rigorous definition of a strange attractor.
Then state the two (2) physical/mathematical features a dynamical system must possess
in order for its flows to be attracted to a strange attractor; write the incompressible N.–S.
equations, and indicate which terms yield the required behaviors.
2
(15) 6. The equations for a k–ε RANS model can be expressed as
∂ ui
− ε + ∇· (ν + νT /σk ) ∇k ,
∂xj
∂ ui
ε2
ε
− Cε2 + ∇· (ν + νT /σε ) ∇ε .
εt + u · ∇ε = Cε1 u0i u0j
k
∂xj
k
kt + u · ∇k = −u0i u0j
(a) Give the physical interpretation of each of the terms (not the individual factors) in
these equations. You need do this for only one of the equations. Explain why.
(b) Use dimensional analysis to obtain a formula for νT in terms of k and ε.
(15) 7. Apply Reynolds averaging to Burgers’ equation given in Prob. 2. Show, using an appropriate definition, that retention of the term ut is incorrect. Then demonstrate how
the Boussinesq hypothesis is employed as a model of the Reynolds stress. Discuss the
mathematical implications of this RANS derivation.
(25) 8. Provide a qualitative, but detailed, discussion in which you compare all of the types of
(“classical”—not synthetic velocity) LES methods described in lectures, namely, Smagorinsky, dynamic (Smagorinsky), scale-similarity, mixed models, and the ILES approach. For
each, discuss the nature of the SGS model (for example, how many constants it contains)
and how it is constructed—no equations necessary, but you may provide them to clarify
your discussions if you wish. In light of the fact that LES is a computational tool, note
the most important function of SGS models.
(20) 9. Show that Burgers’ equation, given in Prob. 2, is Galilean invariant by carrying out the
following steps.
(a) Provide the form of a Galilean transformation.
(b) Give the general form which must be taken by the (transformed) Burgers’ equation
to imply invariance.
(c) Perform the analyses showing that the form in part (b) and the original Burgers’
equation are equivalent.
(d) Suppose this analysis holds for the 3-D N.–S. equations—which it does. Briefly discuss
why this is important. In particular, what must we conclude if it were not true? What
are the implications for typical RANS models, and even many LES procedures?
3