Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: YSYY66
Maths in Action B: Fluid Dynamics
Assignment 1
Upload your report in Gradescope on Learn by 1 Feb at 12 noon . The report should be a single pdf fi le written in LATEX using the template on Learn. The page limit is 3 pages in the format of the template, including fi gures, tables, and references. In addition to the 3 pages, please include all computer codes you used in an appendix (this appendix is not marked). See Learn for details of the marking criteria.
This assignment considers the action on passive scalars of the alternating sine map studied in Workshop 1. The positions of fl uid particles after successive applications of the map is given by the recurrence
xn+1 = xn + u sin(yn + a n), (1a)
yn+1 = yn + u sin(xn+1 + p n), (1b)
where 0 ≤ a n, p n l 2 m are prescribed and u < 0. We examine two cases: (i) the deterministic case, with a n = p n = 0, and (ii) the random case where a n and p n are drawn independently, uniformly in [0, 2 m ).
A passive scalar is released in the fl ow with initial concentration C0( > ) such that
2 m 2 m
C0( > ) d > = 0
0 0
(a constant can be added to the concentration to ensure this holds). The scalar is transported
by the fl ow, which implies that the concentration Cn( > ) after n applications of the map satis fi es
Cn+1( > n+1) = Cn( > n).
Cn+1(x, y) = Cn(x, y ), where y = y − u sin(x + p n), x = x − u sin(y + a n ). (2)
After a few iterations, concentration gradients become large and molecular diffusion acts to smooth them. We model this by replacing (2) by
Cn+1(x, y) =
Cn (x\, y\)e − [(x − x )\2 +(y − y )\2 ]/4 x dx\ dy\, (3)
where the parameter x < 0 captures the effect of molecular diffusion. The model (3) ensures that the total mass of the scalar is conserved:
2 m 2 m 2 m 2 m
Cn( > ) d > = C0( > ) d > = 0.
0 0 0 0
However, for x < 0,
|Cn | = C ( > ) d > n(2) 、 1/2 → 0 as n → 尸 .
Thus the concentration decays to zero everywhere. Physically, the scalar is well mixed, hence homogeneous, as n → 尸. In fact, we expect an exponential decay:
|Cn | s e − n y , i.e. n(l) n − 1 log |Cn | → y < 0, (4)
where y is the so-called decay rate.
The Notebook sinetracer .ipynb implements the map (3) (it uses a Fourier method to compute the convolution integral, but you do not need to worry about this). Using the Note- book as a starting point, investigate the decay of the concentration in both the deterministic and random cases (i) and (ii). Specifically, examine how y depends on x (e.g. for x in the range [10 − 4, 10 − 3] and fi xed u ). Assuming the approximate power-law dependence y ≈ a x b, esti- mate a and b. (Hint: log y ≈ b log x + log a.) Discuss the differences between the random and deterministic cases.