Velocity profiles

In this notebook, we will discuss a useful way to visualize a flow in fluid mechanics: the velocity profile.

Set up the module

using ThermofluidQuantities

using Plots

To start, let's consider one of the most basic types of flow: the linear shear flow, often called Couette flow. This type of flow is generated, for example, between two parallel walls when one wall is moving and the other is stationary. In the figure below, the top wall is moving with velocity $U$, the lower wall is stationary:

Because of the no-slip condition, the fluid next to each wall moves with it. The fluid next to the upper wall moves at velocity $U$, the fluid next to the lower wall is at rest.

The velocity profile depicts $u(y)$, the horizontal component of velocity as a function of vertical position, $y$. This increases linearly from the lower to the upper wall. In fact, the function is just

\[u(y) = Uy/H\]

The arrows indicate the direction that the fluid is moving, and the lengths of the arrows indicate the relative speed at that $y$ position.

Plotting velocity profiles

The arrows are helpful, but you can also plot a velocity profile without them. For example, consider the following velocity:

\[u(y) = \frac{4U_c}{H^2} y (H - y)\]

The coefficient $U_c$ is a speed, and $H$ is the gap height. Let's define a function that evaluates this velocity. Here, y, Uc, and H are to be given as arguments to the function.

u(y,Uc,H) = 4*Uc/H^2*y*(H-y)

u (generic function with 1 method)

Suppose the gap height $H$ is 1 cm and the speed $U_c$ is 1 m/s. We will evaluate this velocity at a range of locations between 0 and $H$:

H = 1u"cm"  # 1 cm = 0.01 m
Uc = 1u"m/s"
y = range(0u"cm",H,length=101) # 101 points to evaluate at, just to make it look smooth.

(0.0:0.01:1.0) cm

Now we evaluate the velocity function at the range of $y$ locations. (Remember that the . vectorizes the evaluation of a function.)

v = Velocity.(u.(y,Uc,H))

101-element Vector{Velocity{Unitful.Quantity{Float64, 𝐋 𝐓^-1, Unitful.FreeUnits{(m, s^-1), 𝐋 𝐓^-1, nothing}}}}:
                 Velocity = 0.0 m s^-1
              Velocity = 0.0396 m s^-1
              Velocity = 0.0784 m s^-1
 Velocity = 0.11639999999999999 m s^-1
              Velocity = 0.1536 m s^-1
                Velocity = 0.19 m s^-1
 Velocity = 0.22559999999999997 m s^-1
              Velocity = 0.2604 m s^-1
              Velocity = 0.2944 m s^-1
              Velocity = 0.3276 m s^-1
                                                                                                                                  ⋮
  Velocity = 0.2943999999999999 m s^-1
 Velocity = 0.26039999999999985 m s^-1
  Velocity = 0.2256000000000002 m s^-1
 Velocity = 0.19000000000000017 m s^-1
 Velocity = 0.15360000000000013 m s^-1
  Velocity = 0.1164000000000001 m s^-1
 Velocity = 0.07840000000000007 m s^-1
 Velocity = 0.03960000000000004 m s^-1
                 Velocity = 0.0 m s^-1

Notice that $u$ is 0 at the beginning and end of the range. Let's plot it. But let's plot it as a velocity profile, which means we make $u$ the 'x' axis and $y$ the 'y' axis.

plot(v,y,xlim=(0u"m/s",2Uc),ylim=(0u"cm",H),
    legend=false,xlabel="u(y)",ylabel="y")

The top and bottom of this plot suggest that these are stationary walls where the flow is at rest. In fact, this is the velocity profile associated with pressure-driven flow through the gap.

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