Consider a flow whose velocity field is given by
u = y,
v = −x
The streamlines are circles centered at the origin, and this flow is an example of a solid-body rotation.
As shown in the figure, a small square fluid element is placed at the point (x,y)=(0,1) at t=0: it moves and distorts with the fluid. Assume that the flow is incompressible.
All of your answers below should use only x, y, Δt, ρ. In the answer boxes, use Dt to denote Δt and rho for ρ.
1) In the first part of this problem, we only consider the fluid element. Determine the angles Δθx and Δθy of the fluid element at some small later time t=Δt.
2) Compute the flow's vorticity field.
ωx=
ωy=
ωz=
3) Compute the rate of strain.
εxx=
εyy=
εxy=
4) Compute the substantial derivatives Du/Dt and Dv/Dt
Du/Dt=
Dv/Dt=
5) For this particular flow, fτi=0. Using the x- and y-momentum equations, find expressions for the components of the pressure gradient ∇p.
∂p∂x=
∂p∂y=
6) Integrate the pressure gradient components to find an expression for p(x,y)−p(0,0).
Is the flow rotational or irrotational?
1) To determine the angles Δθx and Δθy of the fluid element at a small later time Δt, we can use the fact that the velocity field represents a solid-body rotation.
The velocity components u and v give the rates of change of x and y with respect to time. So, we can integrate these velocity components over the time interval Δt to get the changes in x and y coordinates.
Δx = ∫(u) dt = ∫(y) dt = yt + C1
Δy = ∫(v) dt = ∫(-x) dt = -xt + C2
Here, C1 and C2 are constants of integration. Since the fluid element is initially at (x,y) = (0, 1) at t = 0, we can substitute these initial conditions into the equations:
0 = 1(0) + C1 ⇒ C1 = 0
0 = -0(0) + C2 ⇒ C2 = 0
Therefore, the changes in x and y coordinates are:
Δx = yt
Δy = -xt
Now, we can find the angles Δθx and Δθy by taking the arctangent of the changes in y and x, respectively:
Δθx = arctan(Δx / Δt) = arctan(yt / Δt)
Δθy = arctan(Δy / Δt) = arctan(-xt / Δt)
2) The vorticity field (ω) represents the local rotation at a given point in the flow. It is defined as the curl of the velocity field. In this case, the velocity field is:
u = y
v = -x
Therefore, the vorticity components are:
ωx = ∂v / ∂y - ∂u / ∂x = -1 - 1 = -2
ωy = ∂u / ∂x - ∂v / ∂y = 1 - (-1) = 2
ωz = 0 (since the velocity field is 2D)
3) The rate of strain tensor (ε) describes the local deformation of fluid elements. It is defined as the symmetric part of the velocity gradient tensor (∇u). In this case, the velocity gradient components are:
∂u / ∂x = 0
∂u / ∂y = 1
∂v / ∂x = -1
∂v / ∂y = 0
Therefore, the rate of strain components are:
εxx = (∂u / ∂x + ∂u / ∂x) / 2 = 0
εyy = (∂v / ∂y + ∂v / ∂y) / 2 = 0
εxy = (∂u / ∂y + ∂v / ∂x) / 2 = (1 - 1) / 2 = 0
4) The substantial derivatives Du/Dt and Dv/Dt represent the material derivative, which accounts for both advective and convective effects. They can be calculated using the equations:
Du/Dt = ∂u / ∂t + u (∂u / ∂x + ∂v / ∂y)
Dv/Dt = ∂v / ∂t + v (∂u / ∂x + ∂v / ∂y)
In this case, the velocity components are independent of time (∂u / ∂t = ∂v / ∂t = 0), so the substantial derivatives simplify to:
Du/Dt = u (∂u / ∂x + ∂v / ∂y) = y(-1 + 0) = -y
Dv/Dt = v (∂u / ∂x + ∂v / ∂y) = -x(0 + 1) = -x
5) Since fτi = 0, where fτi represents the tangential body force components, we can assume that the flow is inviscid. In an inviscid flow, the pressure is constant along a streamline. Therefore, the pressure gradient components (∇p) can be derived using the x- and y-momentum equations:
x-momentum equation:
ρ (Du/Dt) = - ∂p / ∂x ⇒ ∂p / ∂x = -ρ (Du/Dt) = ρy
y-momentum equation:
ρ (Dv/Dt) = - ∂p / ∂y ⇒ ∂p / ∂y = -ρ (Dv/Dt) = ρx
6) To integrate the pressure gradient components and find an expression for p(x, y) - p(0, 0), we can integrate the expressions derived in the previous step:
∫(∂p / ∂x) dx = ∫(ρy) dx
∫(∂p / ∂y) dy = ∫(ρx) dy
Integrating with respect to x:
p(x, y) - p(0, 0) = ∫(∂p / ∂x) dx = ∫(ρy) dx = ρy²/2 + C1
Integrating with respect to y:
p(x, y) - p(0, 0) = ∫(∂p / ∂y) dy = ∫(ρx) dy = ρx²/2 + C2
So, an expression for p(x, y) - p(0, 0) is:
p(x, y) - p(0, 0) = ρy²/2 + ρx²/2 + C3
To determine whether the flow is rotational or irrotational, we can observe the vorticity field. Since the vorticity components (ωx and ωy) are non-zero in this flow, the flow is rotational.