Fluid Dynamics - Math 6750 Vorticity equation - Week 5 We consider the Navier-Stokes equations for an incompressible Newtonian fluid ∇ · u = 0 (1) ∂u  ρ + u · ∇u = −∇p + ρF + µ∆u (2) ∂t b

1 Bernoulli’s Theorem

Take the body force to be conservative, that is, Fb = −∇χ for some scalar potential χ. Using the vector identity 1 1 u · ∇u = (∇ × u) × u + ∇|u|2 = ω × u + ∇|u|2 2 2 we may rewrite the conservation of linear momentum equation (2) as ∂u 1 1 µ + ω × u = − ∇p − ∇χ − ∇|u|2 + ∆u ∂t ρ 2 ρ or ∂u + ω × u = −∇H + ν∆u, (3) ∂t p 1 2 where H = ρ + χ + 2 |u| and ν = µ/ρ. If the flow is steady, then (3) reduces to ω × u = −∇H + ν∆u. (4) Taking the dot product of (4) against u gives   u · ∇H = −u · (ω × u) + ν∆u = νu · ∆u. (5)

Consequently, H decreases in the flow direction when u·∆u < 0, i.e when the local net viscous force per unit volume tends to decelerate the fluid and work is done against viscous forces as an element of fluid moves along a streamtube; similarly H increases along the streamline when the net viscous force tends to accelerate the fluid. In the case of an ideal fluid (no viscosity), (5) reduces to u · ∇H = 0 which means the directional derivative of H along the flow field u is zero. This is known as Bernoulli streamline theorem. Theorem 1 (Bernoulli streamline thm). For a steady flow of an ideal fluid, subject to a p 1 conservative force F = −∇χ, the function H = + χ + |u|2 is constant along a streamline. b ρ 2 In particular, an increase of the fluid velocity occurs simultaneously with a decrease in the (static) pressure or a decrease in the fluid’s potential energy, along a given streamline. Here, p, χ, u depends on the particular point on the chosen streamline but the constant depends only on that streamline. We point out that the theorem says nothing more than H being constant along a given streamline, so H may have different constants on different streamlines. Recall that a flow is irrotational if ω ≡ 0. In this case, H is constant everywhere

1 Proposition 1. For a steady irrotational flow of an ideal fluid, subject to a conservative p 1 force F = −∇χ, the function H = + χ + |u|2 is constant everywhere. b ρ 2

2 Vorticity Equation

As it turns out, the net viscous force on an element of incompressible fluid is determined by the local gradients of vorticity since

∇ × ω = ∇ × (∇ × u) = ∇ (∇ · u) − ∆u = −∆u. (6)

This is rather surprising since we know from the Newtonian stress tensor that the viscous stress is generated solely by deformation of the fluid and is independent of the local vorticity, but the explanation is wholly a matter of kinematics. The rate of deformation tensor D and the vorticity ω play independent roles in the generation of stress, but certain spatial derivatives of D are identically related to certain derivatives of ω through the vector identity used in (6). It follows from (6) that the viscous distribution of vorticity is pivotal in understanding the evolution of large Reynolds number flow , i.e when the fluid viscosity is sufficiently small. The vorticity transport equation is obtained as follows. Taking the curl of Bernoulli’s equation (3) and using the fact that ∇ × (∇f) = 0 for any C2 scalar function f, we find that

∂ω + ∇ × (ω × u) = ν (∇ × ∆u) . ∂t We use the incompressibility condition and the fact that ∇ · ω = 0 to cancel out terms. Applying the vector identity from (6) to ω reduces the viscous term into

∇ × ∆u = −∇ × (∇ × ω) = −∇ (∇ · ω) + ∆ω = ∆ω.

For the convection term,

∇ × (ω × u) = ω∇ · u − u∇ · ω + u · ∇ω − ω · ∇u = u · ∇ω − ω · ∇u.

Combining all the computations leads to the vorticity equation Dω ∂ω = + u · ∇ω = ω · ∇u + ν∆ω (7) Dt ∂t The pressure term has been eliminated in the vorticity equation, with the price that the vorticity equation contains both u and ω. The terms in (7) can be interpreted: Dω 1. The term is the familiar material derivative of ω, describing the rate of change of Dt ω due to the convection of fluid.

2. The term ν∆ω accounts for the diffusion of vorticity due to the viscous effects.

3. The term ω · ∇u represents the stretching or tilting of vortex tubes due to the folow velocity gradients.

2 4. The term ω (∇ · u) (which is absent due to the incompressibility condition) describes vortex stretching due to flow compressibility.

In the case of a two-dimensional incompressible inviscid flow, i.e u = (u(x, y, t), v(x, y, t), 0), the vorticity only has one non-zero component

ω = (0, 0, ∂xv − ∂yu) = ω(x, y, t)e3 and ∂u ω · ∇u = ω = 0. ∂z Dω Consequently, = 0 and so the vorticity ω of each individual fluid particle does not change Dt in time, i.e if ω = 0 at some time t0, then ω ≡ 0 for all time t ≥ t0. This also occurs for a unidirectional flow, say u = u(y, z, t)e1, since we then have ∂u ∂u ω = (0, ∂ u, −∂ u) =⇒ ω · ∇u = ∂ u − ∂ u = 0. z y z ∂y y ∂z If we further assume steady flow, then Dω = u · ∇ω = 0 Dt and we arrive at the following result

Proposition 2. For a steady two-dimensional flow of an ideal fluid subject to a conservative body force, the vorticity ω is constant along a streamline.

An immediate consequence of this is that steady flow past an aerofoil is generally irrota- tional.

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