An analytical solution of the Boussinesq equations for the motion of a viscous stably stratified fluid driven by a surface thermal forcing with large horizontal gradients (step changes) is obtained. This analytical solution is one of the few available for wall-bounded buoyancy-driven flows. The solution can be used to verify that computer codes for Boussinesq fluid system simulations are free of errors in formulation of wall boundary conditions and to evaluate the relative performances of competing numerical algorithms. Because the solution pertains to flows driven by a surface thermal forcing, one of its main applications may be for testing the no-slip, impermeable wall boundary conditions for the pressure Poisson equation. Examples of such tests are presented.

Thermal disturbances associated with variations in underlying surface properties can drive local circulations in the atmospheric boundary layer (Atkinson, 1981; Briggs, 1988; Hadfield et al., 1991; Segal and Arritt, 1992; Simpson, 1994; Mahrt et al., 1994; Pielke, 2001; McPherson, 2007; Kang et al., 2012) and affect the development of the convective boundary layer (Patton et al., 2005; van Heerwaarden et al., 2014). Computational fluid dynamics (CFD) codes for modelling such flows commonly solve the Boussinesq equations of motion and thermal energy for a viscous/diffusive stably stratified fluid. In this paper we present an analytical solution of the Boussinesq equations for flows driven by a surface thermal forcing with large gradients (step changes) in the horizontal. The solution can be used to verify that CFD codes for Boussinesq fluid system simulations are free of errors, and to evaluate the relative performances of competing numerical algorithms. Such verification procedures are important in the development of CFD models designed for research, operational, and classroom applications.

We solve the linearized Navier–Stokes and thermal energy equations analytically for the case where the surface buoyancy varies laterally as a square wave (Fig. 1). Attention is restricted to the steady state. No boundary-layer approximations are made; the solution is non-hydrostatic, and both horizontal and vertical derivatives are included in the viscous stress and thermal diffusion terms. The solution is similar to that of Axelsen et al. (2010) for katabatic flow above a cold strip but is easier to evaluate (no slope present) and applies to the more general scenario where the viscosity and diffusivity coefficients can differ. The flow is also similar to a special case (no slope) considered by Egger (1981), although a final analytical solution was not provided in that study. Strictly speaking, the linearized Navier–Stokes equations apply to a class of very low Reynolds number motions known as creeping flows. Such flows appear in studies of lubrication, locomotion of microorganisms, lava flow, and flow in porous media. Of course, for the task at hand, if our linear solution is to serve as a benchmark for a nonlinear numerical model solution, it is essential that the parameter space be restricted to values for which the model's nonlinear terms are negligible.

Because the solution pertains to flows driven by a surface thermal forcing, one of its main applications may be as a test for surface boundary conditions in the pressure Poisson equation. In models of atmospheric boundary-layer flows, the buoyancy is a major contributor to the forcing term in the Poisson equation and also appears in the associated surface boundary condition. The pressure boundary condition on a solid boundary in incompressible (Boussinesq) fluid flows is an important and complex issue that has long been fraught with technical difficulties and controversies (Strikwerda, 1984; Orszag et al., 1986; Gresho and Sani, 1987; Gresho, 1990; Temam, 1991; Henshaw, 1994; Petersson, 2001; Sani et al., 2006; Rempfer, 2006; Guermond et al., 2006; Nordström et al., 2007; Shirokoff and Rosales, 2011; Hosseini and Feng, 2011; Vreman, 2014). Typical fractional-step solution methodologies and associated pressure (or pseudo-pressure) boundary-condition implementations are often verified using various prototypic flows such as Poiseuille flows, lid-driven cavity flows, flows over cylinders or bluff bodies, viscously decaying vortices, and dam-break flows. We are unaware of verification tests in which flows were driven by a heterogeneous surface buoyancy forcing. Our solution is designed to fill this gap.

Schematic of two-dimensional (

The analytical solution is derived in Sect. 2. In Sect. 3, this solution is compared to numerically simulated fields in a steady state. Two versions of a numerical code are run: a version in which the correct surface pressure boundary condition is applied, and a version in which the pressure condition is mis-specified. A summary follows in Sect. 4.

We derive the solution for steady flow over an underlying surface along which the buoyancy varies laterally as a single-harmonic function. This single-harmonic solution is then used as a building block in a Fourier representation of the square-wave solution.

Consider the flow of a viscous stably stratified fluid that fills the
semi-infinite domain above a solid horizontal surface (placed at

We obtain our solution using a standard vorticity/streamfunction
formulation. Cross-differentiating Eqs. (1) and (2) yields the vorticity
equation,

For a surface buoyancy of the form

With the general solution for

The pressure follows from Eqs. (1) and (12) as

The surface conditions determine

Next, consider the case where the surface buoyancy varies horizontally as a
square wave, with a distribution over one period

A solution of the linearized equations may be used to verify a nonlinear
code if the nonlinear terms are sufficiently small. Unfortunately, a priori
estimates of such terms expressed, for example, through a Reynolds number,
are not straightforward since the relevant velocity and length scales in our
problem are only evident after a solution has been obtained. We thus seek an
appropriate set of test parameters through trial and error, guided by a posteriori
linear solution estimates of the terms

Vertical cross section of the analytical (A-1) buoyancy

Vertical cross section of

Vertical cross section of the analytical (A-2) buoyancy

Vertical cross section of

Vertical cross section of

The numerical model employed in our tests is a variant of a direct numerical
simulation (DNS) code used in the boundary-layer and slope-flow studies of
Fedorovich et al. (2001), Fedorovich and Shapiro (2009a, b), and Shapiro and
Fedorovich (2013, 2014). The model solves the Boussinesq governing equations
on a staggered (Arakawa C) grid. Although designed for three-dimensional
simulations, the model was run in a two-dimensional (

The analytical solution was evaluated on an un-staggered (

In the first test, we set

To understand why the INC-1 and HNC-1 simulations are so similar, and to
identify simulation parameters that might evince more substantial
differences, we consider the idealized problem in which a specified buoyancy

The preceding analysis suggests that simulations with shallow thermal
disturbances (

In the second test, we set

The linearized Boussinesq equations for the motion of a viscous stably stratified fluid are solved analytically for a surface buoyancy that varies laterally as a square wave. The solution describes two-dimensional laminar convective structures such as thermal convective rolls and updraft/downdraft pairs. The main applications of the solution may be in code verification and the evaluation of different implementations of the surface pressure condition for the pressure Poisson equation. Tests have been conducted for cases where the aspect ratios of the thermal disturbance have been large and small. With attention restricted to disturbances of sufficiently small amplitude, the linear solution and numerically simulated fields with the inhomogeneous Neumann condition for pressure (which is appropriate in the context of the particular fractional step procedure adopted in our DNS code) have been found to be in excellent agreement for both tests. However, in tests with a mis-specified Neumann condition, an excellent agreement with the analytical solution has been found only for the deep (small aspect ratio) disturbance case; errors in the shallow (large aspect ratio) disturbance case have been catastrophic.

Consider a three-dimensional Boussinesq system with the equation of motion

The same steps leading to Eq. (A3a) also lead to an alternative Poisson
equation,

Evaluating the vertical component of Eq. (A1) on the surface, where the
impermeability condition applies, yields the inhomogeneous Neumann
condition,

In our numerical model, Eq. (A1) is integrated using a fractional step procedure
with explicit treatment of the viscous term. First, a provisional velocity
field

In some explicit fractional step procedures (including the DNS code used in
our study), the problem of solving Eq. (A5) subject to Eq. (A6) with

Finally, we note that in fractional step procedures that treat the viscous
term implicitly (e.g. Kim and Moin, 1985; Gresho, 1990; Armfield and
Street, 2002; Guermond et al., 2006, and many others), the homogeneous
Neumann condition is often applied as a surface condition for a Poisson
equation, but it is again different from our implementation described in
Sect. 3. In the implicit treatments, the provisional velocity is obtained as
the solution of a boundary value problem (

The Fortran program used to generate output data files from the analytical
solution is available as a supplement to this article. That program
(square.f) is configured for test A-1 but can be easily adjusted to run
test A-2 or other tests. Running square.f automatically generates an output
file for each dependent variable (e.g. u.dat) as well as an output file
(square.out) that summarizes the test parameters and gives the computed
values of the linearity ratios

This research was supported by the National Science Foundation under grant AGS-1359698. Comments by Chiel van Heerwaarden, Juan Pedro Mellado, Inanc Senocak, and an anonymous reviewer are gratefully acknowledged. Edited by: D. Lawrence