8.7 Nonuniform inlet velocity

At an inlet boundary, the velocity is usually specified by a fixed value condition eqn, as discussed in Sec. 4.3 . In many CFD applications, the flow at an inlet boundary is described by a single speed which is applied to all faces assuming eqn is uniform across the boundary.

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This creates an anomaly where the inlet boundary meets a wall. In the vicinity of the two boundaries, the flow decelerates from eqn at an inlet face to a value in the adjacent cell close to the no-slip condition eqn applied at the wall.

There is inevitably a high “spike” in pressure and shear stress within the cell in order to decelerate the flow so rapidly. As the length of the cell (in the flow direction) is reduced, the deceleration and associated pressure eqn increases such that eqn in the limit that cell volume eqn.

The solution tends to converge more slowly with the pressure spike, and can be unstable. Furthermore, the spike in shear stress can generate high levels of turbulence which can cause flow separation where the inlet and wall meet.

The uniform condition does not reflect the flow behaviour upstream of the inlet. For example, assuming the wall extends upstream, a boundary layer would have developed at the inlet; or if the wall begins at the inlet, the flow would stagnate at its leading edge.

This is a good example of the axiom that numerical methods do not respond well to any modelling which is unphysical. The problems can be avoided by specifying a nonuniform eqn which represents the upstream flow better.

Some fields of engineering use established theories to describe the inlet eqn, e.g. wind engineering uses a profile of eqn based on an atmospheric boundary layer along the earth’s surface.

More generally, a nonuniform eqn can be specified which tends to eqn at wall boundaries. Profiles can be described by eqn where: eqn is the normalised velocity and eqn is normalised distance to the nearest wall boundary; and, eqn and eqn denote the maximum eqn and eqn values.

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It is logical to specify eqn using established profiles for boundary layers as a reasonable estimate for the upstream conditions. A quadratic function eqn represents a developed boundary layer for laminar flow, matching the analytical profiles for flow in a pipe or between flat parallel plates, i.e. Poiseuille’s law.

Alternatively, a power law function eqn represents a developed turbulent boundary layer quite well. Prandtl used eqn — his one-seventh power law3 — to reproduce data for flow in a pipe, but any suitable exponent eqn can be used in practice.

3Ludwig Prandtl, The mechanics of viscous fluids, 1935.

Notes on CFD: General Principles - 8.7 Nonuniform inlet velocity