Notes on Computational Fluid Dynamics: General Principles

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8.7 Nonuniform inlet velocity

At an inlet boundary, the velocity is usually speciﬁed by a ﬁxed value condition $\text{[math]}$ , 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 $\text{[math]}$ 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 $\text{[math]}$ at an inlet face to a value in the adjacent cell close to the no-slip condition $\text{[math]}$ 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 $\text{[math]}$ increases such that $\text{[math]}$ in the limit that cell volume $\text{[math]}$ .

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 $\text{[math]}$ which represents the upstream ﬂow better.

Some ﬁelds of engineering use established theories to describe the inlet $\text{[math]}$ , e.g. wind engineering uses a proﬁle of $\text{[math]}$ based on an atmospheric boundary layer along the earth’s surface.

More generally, a nonuniform $\text{[math]}$ can be speciﬁed which tends to $\text{[math]}$ at wall boundaries. Proﬁles can be described by $\text{[math]}$ where: $\text{[math]}$ is the normalised velocity and $\text{[math]}$ is normalised distance to the nearest wall boundary; and, $\text{[math]}$ and $\text{[math]}$ denote the maximum $\text{[math]}$ and $\text{[math]}$ values.

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It is logical to specify $\text{[math]}$ using established proﬁles for boundary layers as a reasonable estimate for the upstream conditions. A quadratic function $\text{[math]}$ 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 $\text{[math]}$ represents a developed turbulent boundary layer quite well. Prandtl used $\text{[math]}$ — his one-seventh power law³ — to reproduce data for ﬂow in a pipe, but any suitable exponent $\text{[math]}$ can be used in practice.

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

Notes on CFD: General Principles - 8.7 Nonuniform inlet velocity