Sebastian Henao – Thermal Engineering Course – NYU Tandon School of Engineering
Analysis of Unsteady Couette Flow by Computational Fluid Dynamics Methods
Introduction
Computational fluid dynamics (CFD) is one of the branches of study of fluid mechanics that uses numerical
methods to solve and analyze problems related to fluid flows. In this project we analyze two problems
numerically worked out by writing a computer code in MATLAB involving simple partial differential
equations (PDEs) with specified boundary/initial conditions. In first place is presented the unsteady
Couette Flow, which is related to the movement of a viscous fluid between two surfaces, one of which is
moving tangentially relative to the other.
A. Assessment of the behavior of the error E1 and E2 as a function of both spatial and temporal
resolution.
One of the physical interpretations of the Couette flow equation is the rate at which the velocity at a given
point changes over time, depends on the second derivative of that velocity at that point with respect to
space, that is, where there's curvature in space there's change in time.
The PDE only describes one out of three constraints that the velocity function must satisfy, but for
describing accurately this flow velocity, it must also satisfy certain boundary and initial conditions. Joseph
Fourier solved this problem in 1822, his key contribution was to gain control of the solution, by breaking
down into three observations
1. The number of certain sine waves offer a simple solution to this equation
2. Linearity: If you know multiple solutions, the sum of these functions is also a solution
3. Fourier Series: The Function can be expressed as a sum of sine waves
Focusing on the first point, let approach in a more general way the velocity to the function 𝑢(𝑦, 0) =
sin(𝑦), where 𝑦 describes a point between the two plaquettes. The right-hand side of this Couette
equation
𝜕𝑢
𝜕𝑡
𝜕2 𝑢
= 𝜈 𝜕𝑦2 (0) asks about the second derivative of this general function, on how much the
velocity distribution curves as it moves along space:
𝜕𝑢
(𝑦, 0) = cos(𝑦) (1)
𝜕𝑦
𝜕2𝑢
(𝑦, 0) = −sin(𝑦) (2)
𝜕𝑦 2
As seen on the last derivation (1) and (2), the amount the wave curves are equal and opposite to its height
at each point, so at the point where time 𝑡 equals zero, it is observed that each point changes it’s velocity
at a rate proportional to the to the velocity itself, with the same proportionality along all points:
𝝏𝒖
(𝑦, 0)
𝝏𝒕
𝜕2 𝑢
= 𝜈 𝜕𝑦2 = −𝜶 . 𝒖(𝑦, 0), consider 𝜶 as the constant of proportionality
So, after some tiny time steps, everything scales down by the same factor, and after that it’s still the same
sine curve shape just scaled down a bit uniformly, and this happens as well in the limit as the size of the
time steps approaches zero
𝒖(𝑦, ∆𝑡) = 𝑐. sin (𝑦)
𝒖(𝑦, 2∆𝑡) = 𝑐 2 . sin (𝑦)
𝒖(𝑦, 3∆𝑡) = 𝑐 3 . sin (𝑦)
𝒖(𝑦, 4∆𝑡) = 𝑐 4 . sin (𝑦)
Now, looking that the rate at which some values changes is proportional to that value itself, it seems
indeed similar of an exponential, where we can take as an example:
𝑑
𝐶𝑒 −0.003𝑡 = −0.003. 𝐶𝑒 −0.003𝑡 (3)
𝑑𝑡
What can be said of this equation (3), is that the derivative of 𝑒 to some constant times 𝑡, is equal to that
constant times itself, so when we look at the at the Couette equation (0) and it is known that for a sine
wave the right-hand side is going to be a negative alpha times the time the velocity function itself, it is not
surprising to propose that the solution scales down by a factor of 𝑒 to the negative 𝜶𝒕.
𝝏𝒖
= = −𝜶 sin(𝑦) . 𝑒 −𝜶𝒕
𝝏𝒕
𝟐
As seen on the figure 1 below for the exact solution equation 𝑢𝑒 (𝑦, 𝑡) = 𝑦 + 𝑒 −𝝅 𝒕 sin (𝜋𝑦) , the larger
the time 𝑡 gets (trending to infinity), the smaller the exponential function becomes, trending therefore
to zero the exponential portion, and making the exact solution 𝑢𝑒 (𝑦, ∞) a value more similar to the
result of the grid point location 𝑢𝑒 (𝑦, ∞)~ 𝑦𝑗 where, 𝑦𝑗 = (𝑗 − 1)Δ𝑦 for 𝑗 = 1, 2, … 𝑁
Figure 1. 3D graph for the exact solution equation of the Unsteady Couette Flow
That means that the slope of the exponential function tends to zero as it advances in time, even for the
𝑦
𝜋𝑦
boundary and initial conditions described as 𝑢(0, 𝑡) = 0, 𝑢(ℎ, 𝑡) = 𝑈 and 𝑢(𝑦, 0) = 𝑈 ℎ + 𝑈𝑠𝑖𝑛( ℎ ),
which can be mathematically presented as
𝜕𝑢
(0, 𝑡)
𝜕𝑦
=
𝜕𝑢
(𝑦, 𝑡)
𝜕𝑦
= 0.
Another aspect to note from the exact solution equation is that the sine function sin (𝜋𝑦) frequency is
affected by the terms multiplied inside by the input “y” of the function, in this case “π”, since a higher
frequency waves curve more sharply and lower frequency one’s more gently. That means that by
introducing some constant multiplied by the input “y”, the function oscillates more quickly. If for instance,
we take the derivative of
𝜕
cos(𝜋𝑦)
𝜕𝑦
= −𝜋 sin(𝜋𝑦) and similarly the second derivative would be still
−𝜋 2 cos (𝜋𝑦), what this means for this velocity function, is that it decays more quickly towards and reach
equilibrium over time.
B. Assessment of the solution behavior as a function of time step (∆t) and number of nodes (N)
The velocity is calculated in steps of time, using the inverse of the matrix A multiplied by the vector b,
from equation 𝐴𝑥 = 𝑏. The results for the velocity profiles are shown from the stages in the time marching
process are shown in the figure below. One of the most noticeable facts at first sight is that the velocity is
changing most rapidly near the upper plate, was expected.
Figure 2. Unsteady Couette flow velocity profiles in the time-stepping process
The influence of the shear stress that comes from the upper plate is increasingly circulated to the rest of
the fluid, resulting in a final, steady-state profile. The steady state profile is linear, agreeing with the
analytical exact solution. To provide a more direct comparison of your computations with the present
calculations.
In terms of stability, it is observed that the behavior of the solution is not compromised, as the CrankNicolson technique is unconditionally stable. When ∆t is increased (and letting fixed the values of
kinematic viscosity and the number N of nodes), the accuracy is affected, and the number of marching
steps required to obtain a steady state change
j
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
y/D
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000
Δt = 0.001
0
1.51E-09
5.79E-09
2.05E-08
7.16E-08
2.47E-07
8.42E-07
2.83E-06
9.39E-06
3.06E-05
9.80E-05
0.000307
0.000935
0.002765
0.007891
0.021552
0.055767
0.134806
0.298529
0.58928
1
Δt = 0.003
0
1.52E-05
3.82E-05
8.07E-05
0.0001637
0.0003263
0.000643
0.0012527
0.002411
0.0045781
0.0085627
0.0157451
0.0283997
0.0501113
0.0862183
0.1440806
0.2327597
0.3614702
0.5361487
0.7542414
1
𝑈 𝑈𝑒
Δt = 0.01
0
0.0031622
0.0066738
0.0109121
0.0163107
0.0233884
0.0327785
0.0452577
0.0617703
0.0834451
0.1115974
0.147707
0.1933615
0.2501531
0.3195181
0.4025174
0.4995666
0.6101501
0.7325804
0.8638961
1
Δt = 0.03
0
0.02463
0.0497626
0.0758973
0.1035279
0.133138
0.165195
0.2001435
0.2383951
0.2803175
0.3262199
0.3763381
0.4308172
0.4896942
0.5528822
0.6201571
0.6911487
0.7653402
0.8420767
0.9205839
1
Δt = 0.1
0
0.046808
0.093691
0.140726
0.187983
0.235529
0.283423
0.331718
0.380457
0.429671
0.479384
0.529604
0.580328
0.63154
0.683214
0.735309
0.787774
0.840548
0.893562
0.946739
1
Table 1. Transient velocity profiles comparison
Five profiles are given, for ∆t = 0.001, 0.003, 0.01, 0.03, 0.1, these are transient profiles, all corresponding
to the same nondimensional intermediate time t = 5. After examining these results, is evident that the
most similar columns are ∆t = 0.001 and ∆t = 0.003. Since both are small time steps, it can be assured that
with a value as big a ∆t = 0.003 it provides a timewise accuracy for the implicit calculations.
However, looking the last three columns for ∆t = 0.01, 0.03, 0.1. There are some differences between
these results and ∆t = 0.001, especially comparing them near the upper wall. Apparently, ∆t = 0.01
corresponds to a large-enough value of ∆t to cause some noticeable inaccuracy for the transient results.
This inaccuracy continues to grow as ∆t is further increased. And it is even more evident if has a large
value such as ∆t=1 for example. Here, the value of ∆t is so large that not timewise accuracy can be
achieved, and none is obtained. Exhibiting nonphysical behavior, especially near the top plate.
From table 2, it can also be observed the total time steps required for each time step value, for instance
using a ∆t = 0.003, the steady state is obtained after 538-time steps. However, for large values of ∆t, as
0.01 and 0.03, about 167 and 61 steps are required respectively, as seen on the table below:
Non-dimensional time step Δt Maximum iteration for convergence
𝑡
0.001
0.003
538
0.01
167
0.03
61
0.1
23
Table 2. Convergence time for different time steps
dy
jmax
0.1
0.05
0.025
0.0125
11
21
41
81
Time Steps
(Δt = 0.003)
382
380
380
380
Max RMS Error 1
(Δt = 0.003)
7.62E-01
7.79E-01
7.84E-01
7.86E-01
Time Steps
(Δt = 0.03)
43
43
43
43
Max RMS Error 1
(Δt = 0.03)
5.19E-01
5.22E-01
5.23E-01
5.23E-01
Table 3. Comparison of maximum RMS error as a function of different time steps
From table 3, the program has been iterated until reaching an “error 2”, expressed as 𝐸2 < 𝜖 = 1𝑥10−5.
Each iteration has been done changing the number of nodes “N” in the “yj” direction from 11 to 81, with
two different time steps ∆t = 0.003 and ∆t = 0.03. The purpose of this calculation was to see the difference
between each maximum RMS Error 1 value reached and the number of time steps that took place for the
convergence of the code. Observing the results from the time step ∆t = 0.003, the total number of time
steps remained invariable no matter how much nodes were applied for the iteration. In the other hand it
is observed a trend, taking a glace in N=11 the Max RMS Error 1 was 7.62E-01, a value that presented
positive variation as the value of the total nodes increased, in this case we end up having a maximum RMS
of 7.86E-01 for the 81 nodes of the last computing. This same behavior is observed also for the 10 times
greater time step value of ∆t = 0.03, where the total number of time steps remained the same, but the
Maximum RMS Error 1 value increased from 5.19E-01 to 5.23E-01
C. Conclusions:
1.Time accuracy is adrift when a greater ∆t is used for the calculations. It can be concluded from these
results that implicit methods, such the one of Crank-Nicholson, with large values of ∆t are not the methods
to use for problems wherein the flows that are changing from one steady to another steady state. In this
study case, time accuracy can be obtained for smaller values of ∆t but needing more steps to meet the
steady state.
2. Increasing ∆t, a reduction in the number of time steps needed to obtain a steady state is observed.
However, for a large-enough value of ∆t, the trend reverses itself, and the value of using an implicit
method is completely lost. Therefore, there is an optimum value of ∆t which leads allows to implement
the most efficient of the Crank-Nicolson method in a case by case.
3. Increasing the number of nodes in the solution of Crank-Nicholson method, has no effect on the number
of time steps required to implement the computing of the program. However, the accuracy of the solution
could be compromised, as the RMS error trend to grow with larger number of nodes, therefore requiring
a smaller number of time step ∆t to maintain a reliable calculation