Models of Flow Separation

The holy grail of CFD as computational fluid mechanics is:

• Turbulence modeling.
• Flow separation.

DFS as Direct Finite Element Simulation offers answers to these problems:

• Turbulence captured as best possible computational solution to the Euler equations.
• Flow separation described as 3d rotational or parallel slip separation.  

We here give the elements of separation in bluff body flow as illustrated in the pictures above of an airplane landing gear (with further details in New Theory of Flight and Secret of Flight):

1. 3d rotational slip with point stagnation (back and side of wheels).
2. 3d parallel slip with 2d line stagnation (top of wheel support). 

We start from the following basic observations:

• Separation in 2d potential flow can only take place a stagnation with zero flow velocity.
• Accelerating flow is stable in velocity and unstable in vorticity.
• Decelerating flow is unstable in velocity and stable in vorticity.
• Rotational flow is neutrally stable. 

We consider 2d potential flow in a $(x_1,x_2,x_3)$ coordinate system around a long cylinder with axis in the $x_3$-direction and flow in the $x_1$-direction, in the back modeled by the velocity    

• $u(x)=(x_1,-x_2,0)$ in the half-plane $\{x_1>0\}$                                              (1)

We observe the critical element of separation away from the plane $\{x_1=0\}$ representing the back surface of the body, through the positive velocity $u_1=x_1$, which is balanced to maintain incompressibility by the opposing flow $u_2=-x_2$, with 2d stagnation with $x_1=x_2=0$ along the $x_3$-axis. We recall that opposing flow is unstable in 3d and thus $u_2=-x_2$ generates rotational flow from a perturbation oscillating in the $x_3$-direction:   

• $u(x)=(0,x_3,-x_2)$ in the half-plane $\{x_1>0\}$

as counter-rotating tubes of stream-wise vorticity in the $x_1$-direction attaching to the plane $\{x_1=0\}$. This leads to a combined quasi-stable separation pattern of the form  

• $u(x)=(2\epsilon x_1,x_3-\epsilon x_2,-x_2-\epsilon x_3)$ in the half-plane $\{x_1>0\}$ (2)

with some $\epsilon \gt 0$, which is characterised as rotational flow with 3d point stagnation as seen in the oil film visualisation above, in the rotational flow in a bath-tub drain and in the rotational rising (separating) flow of a tornado. Instability of potential flow with separation from 2d line stagnation is thus turned in 3d quasi-stable rotational separation from 3d point stagnation. The flow accelerating in the $x_1$-direction is stable in velocity, but unstable in stream-wise vorticity which intensifies the swirling motion into turbulence (vortex stretching).

The oil film picture also shows parallel separation from lines of converging flow lines with transversal stagnation superimposed on a main flow, which we model by the velocity   

• $u(x)=(1,x_2,-x_3)$ in the half-plane $\{x_2>0\}$,                                (3)

with flow separating from the surface $\{x_2>0\}$ with velocity $u_2=x_2$, balanced by the opposing flow $u_3=-x_3$. In this case the instability of opposing flow potentially generating vorticity in the $x_2$-direction, is "swept" away by the main flow $u_1=1$.  

The vortical flow in (3) with the $x_2=0$-plane as the upper surface of a wing can be seen to be generated by vortex stretching in accelerating flow on the leading edge. In this case the stabilisation from the main flow may be insufficient, which may lead to 3d rotational slip separation into the half space $\{x_2>0\}$ and then connects to stall. This phenomenon is also seen on the inner side of the wheels above.

We can thus summarise quasi-stable patterns of flow separation with slip as: 

• 3d rotational with point stagnation modeled by (2). (back of wheel)
• Parallel with 2d line stagnation modeled by(3). (top of wheel support)
• Parallel 3d rotational modeled by (3) + properly modified form of (2). (inner side of wheel)  

In the standard boundary layer theory with no-slip by Prandtl, flow separation is connected to stagnation from adverse pressure gradient. With this theory flow separation has remained a mystery. As a consequence CFD with no-slip following Prandtl does not truely capture flow separation.

In short: DFS offers a resolution to the two main open problems of CFD: turbulence and flow separation. DFS also opens to theoretical understanding for the first time of the complex phenomenon of partly turbulent bluff body flow, which is captured in the following mantra:

• bluff body flow = potential flow modified by 3d rotational or parallel slip separation. 

We understand that flow separation in potential flow is unstable, while flow attachment is more stable because the opposing flow is not present. This is what makes bluff body flow largely stay potential until separation, as seen on the outside of the wheels. We see that flow separation is a large scale phenomenon and that turbulence arises in the vortical swirling flow after separation.