Dirichlet boundary conditions prescribe the numerical value that the variable at the domain boundary should assume when solving the governing ordinary differential equation (ODE) or partial differential equation (PDE).
Solving a Dirichlet problem requires finding the function of the governing partial differential equation at the variable specified domain boundary.
An understanding of the various boundary conditions helps systems designers analyze the accuracy of their fluid system simulation.
An example of Dirichlet boundary conditions in fluid system analysis
The flow dynamics near the boundary of a fluid system have their own unique attributes, and reflecting this on a simulation domain is challenging. An understanding of these different boundary conditions enables designers to design fluid systems that represent optimal real-world flow behavior, either in aerodynamic or hydrodynamic applications.
During CFD analysis, designers often come across two major types of boundary conditions: Dirichlet and Neumann boundary conditions. Dirichlet boundary conditions assume the solution to the variable. In Neumann boundary conditions, a solution is assumed for the derivative of the variable. In this article, we will focus our discussion on Dirichlet boundary conditions and their role in solving fluid dynamics problems.
What Are Dirichlet Boundary Conditions?
Dirichlet boundary conditions, also referred to as first-type boundary conditions, prescribe the numerical value that the variable at the domain boundary should assume when solving the governing ordinary differential equation (ODE) or partial differential equation (PDE). In fluid dynamics, it is usually the pressure or velocity values that are imposed on the boundary of the domain. For instance, when solving fluid dynamics problems for a viscous fluid with no-slip and no-penetration conditions, according to the Dirichlet boundary condition, the component of velocity should be zero at the boundary.
When solving the ordinary differential equation, y”+y=0, the Dirichlet boundary condition for the interval (a,b) can be expressed as y(a) =α and y(b)=β, where α and β are the fixed given numbers.
However, when solving the partial differential equation, the boundary condition can be specified as:
Note that ▽² is the Laplace operator.
In systems that are not bounded, there is no need to define a boundary condition. However, considering most systems in nature have boundaries, the Dirichlet equation for a bound domain Ω⊂Rⁿ can be expressed as:
Note that f is the prescribed function on the boundary ∂Ω.
Comparing Dirichlet and Neumann Boundary Conditions
Let’s take the above partial differential equation, for instance, ▽²y+y=0.
When solving the boundary problem, the Neumann boundary condition assumes the value of the derivative of the variable applied at the domain boundary, as opposed to the variable itself as in the Dirichlet boundary condition.
Thus, for a domain condition Ω⊂Rⁿ, the Neumann boundary condition can be expressed as:
Solving the Dirichlet Problem
The Dirichlet problem is primarily used to solve partial differential equations in heat transfer or fluid flow problems, where a value can be specified to the boundary of the domain. For domain D with boundary ∂D, the Dirichlet problem can be expressed as:
Note that G(x,s) is Green’s function for a specified boundary condition.
Using the Fredholm equation of the second kind, the above function can be expressed as:
For s∈ ∂D and x∈ D, Green’s function for the above equation is G(x,s) = 0. This function provides the harmonic solution to the governing partial differential equation.
Analyze Boundary Conditions With CFD Solvers
CFD solvers allow systems designers to ideally define flow regions and analyze the Dirichlet boundary conditions of a simulation. Whether in laminar or turbulent flow regions, the governing equations can be discretized along the boundary of the domain to find the solution for the Dirichlet problem. This can be facilitated by platforms such as Omnis, which uses FEA and CFD methods for accurate simulation modeling for either compressible or incompressible flow.