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Barotropic Kelvin and Bernoulli with exact hypotheses — worked interpretation

Physical scope

Constructed exact barotropic Euler field with specified parameters; not an empirical claim about viscous fluids.

Example

Choose $P (ρ) = 4 m^{2} s^{- 2} ρ$ , $Ω = 1 s^{- 1}$ , $ρ_{c} = ρ_{r e f} = 1 k g m^{- 3}$ , and an open cylinder with radius $R = 2 m$ , half-height $H = 1 m$ . The exact steady density is $ρ (r) = e^{r^{2} / (8 m^{2})} k g m^{- 3}$ and $u = Ω (- y, x, 0)$ , with $x, y$ in metres. On a material circle of radius $1 m$ , $Γ = 2 π m^{2} s^{- 1}$ . Between $r = 0$ and $r = 1 m$ , Bernoulli value changes by $1 m^{2} s^{- 2}$ ; each circular streamline still has constant value.

Example references

Mandre, Fluid Dynamics Lecture Notes, Chapter 5, §§5.2.1–5.2.2, incompressible Bernoulli context only; the chapter does not treat the barotropic/compressible result used here

Facts & Assumptions

Given: $P = c^{2} ρ$ with $c = 2 m s^{- 1}$ , $Ω = 1 s^{- 1}$ , $ρ_{c} = ρ_{r e f} = 1 k g m^{- 3}$ , and the exact solid-rotation field.

[F1]

Barotropic Kelvin and Bernoulli with exact hypotheses — conditional physical prediction: circulation is $2 π Ω s^{2}$ and Bernoulli is a constant plus $Ω^{2} r^{2}$ in this smooth barotropic Euler model.

[F2]

Formal parameterized fluid systems and observables: for a closed piecewise- $C^{1}$ oriented curve, circulation is $\oint_{γ} u \cdot d r = \int u (γ (s)) \cdot γ^{'} (s) d s$ . Here the material circle in the $z = 0$ plane is $X (t, θ) = ((1 m) \cos (θ + Ω t), (1 m) \sin (θ + Ω t), 0)$ for $t \in R$ seconds and $0 \leq θ \leq 2 π$ , with $Ω = 1 s^{- 1}$ ; $\partial_{t} X = u (t, X)$ and $Γ (t) = \int_{0}^{2 π} u (t, X (t, θ)) \cdot \partial_{θ} X (t, θ) d θ$ .

Proof

1.1givenF1F2algebra

For the radius-one material circle $X$ in [F2], its circulation is $Γ (t) = \int_{0}^{2 π} u (t, X (t, θ)) \cdot \partial_{θ} X (t, θ) d θ = 2 π Ω (1 m)^{2} = 2 π m^{2} s^{- 1}$ .

2.1givenF1step 1.1algebra∎

The Bernoulli difference from $r = 0$ to $r = 1 m$ is $1^{2} (1^{2} - 0^{2}) = 1 m^{2} s^{- 2}$ . It is a between-streamline difference; each circle keeps fixed $r$ .

Depends on

Dependency tree · 2 levels

Sources

Mandre, Fluid Dynamics Lecture Notes