Tensors and Navier Stokes Equations

More Notes for CFD Class

Tensors

Tensors resemble matrices. They contain more information than vectors. A tensor is created when taking the gradient of a vector so they will show up soon when we want the velocity gradient for shear stress.

An example is the Cauchy stress tensor which is constructed partially from shear stresses which are related to velocity gradient.

Taking the gradient of velocity in 3D cartesian coordinates returns a tensor.

Dyadic Product and Tensor Operations

The Dyadic product of 2 vectors generates a tensor. Using convective flux as an example where q is a vector quantity this time.

The divergence of a tensor is a vector.

As previously shown, the gradient of a vector is a tensor.

Continuity Equations

The integral form of the continuity equation was previously shown. The source term is zero and the only form of flux is convective. The differential form can also be created. The convective flux can be broken up to create the advective form which is something…

Momentum Equation

The momentum equation is an expression of vector conservation since momentum is a vector quantity. Helpful to compare to F=ma.

Convective flux a tensor since momentum is a vector. Diffusive flux is from Cauchy stress tensor.

The Cauchy stress tensor is a function of pressure stress and viscous stress. Viscous stress is a function of viscosity and gradients of velocity. Viscous stress can be neglected either if the viscosity is negligibly low or if the flow is super uniform and velocity gradients are negligible.