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3.7 Crocco’s Equation

Chalmers University of Technology
Department of Mechanical Engineering
Division of Fluid Dynamics

The momentum equation without body forces

ρDvDt=−∇p\rho\frac{D\mathbf{v}}{Dt}=-\nabla p
(3.88)

Expanding the substantial derivative

ρ∂v∂t+ρv⋅∇v=−∇p\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p
(3.89)

The first and second law of thermodynamics gives

T∇s=∇h−∇pρT\nabla s =\nabla h-\frac{\nabla p}{\rho}
(3.90)

Insert ∇p\nabla p from the momentum equation

T∇s=∇h+∂v∂t+v⋅∇vT\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}
(3.91)

Definition of total enthalpy (hoh_o)

ho=h+12v⋅v⇒∇h=∇ho−∇(12v⋅v)h_o=h+\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\Rightarrow \nabla h=\nabla h_o-\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)
(3.92)

The last term can be rewritten as

∇(12v⋅v)=v×(∇×v)+v⋅∇v\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)=\mathbf{v}\times(\nabla\times\mathbf{v})+\mathbf{v}\cdot\nabla\mathbf{v}
(3.93)

which gives

∇h=∇ho−v×(∇×v)−v⋅∇v\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}
(3.94)

Insert ∇h\nabla h in the entropy equation gives

T∇s=∇ho−v×(∇×v)−v⋅∇v+∂v∂t+v⋅∇vT\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}
(3.95)
T∇s=∇ho−v×(∇×v)+∂v∂tT\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}
(3.96)
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