Artificial Pressure Method for Solving the Navier-Stokes Equations

Overview

This project implements a finite volume method to solve the Navier-Stokes equations using an artificial pressure approach. The method is applied to the cavity benchmark problem and forward-step benchmark problem. Artificial pressure is introduced to address the decoupling between pressure and velocity fields, prevent pressure oscillations, and allow solving the equations on the same grid without the need for a staggered grid.

Methodology

The artificial pressure approach modifies the governing equations to maintain stability and ensure a coupled solution for velocity and pressure. A structured Cartesian grid is used, and boundary conditions are enforced at the domain edges. For incorporating solid geometries like a forward-step, a mask function is implemented to distinguish fluid and solid regions. The mask assigns a value of 1 to fluid regions and 0 to solid regions, ensuring that computations are performed only in the fluid domain.

Mask Function Implementation

A mask matrix is used to represent the domain, where fluid regions have a value of 1 and solid regions have a value of 0. The function mask_fn initializes this mask and ensures computations are skipped for solid regions:

def mask_fn(x0_s, x1_s, y0_s, y1_s, u, dx, dy):
    mask = np.ones(u.shape)
    L_x = x1_s - x0_s
    L_y = y1_s - y0_s
    n_start_step_x = round((x0_s - 0) / dx)
    n_start_step_y = round((y0_s - 0) / dy)
    n_end_step_x = round(n_start_step_x + (L_x) / dx)
    n_end_step_y = round(n_start_step_y + (L_y) / dy)

    for i in range(n_start_step_y + 1, n_end_step_y + 1):
        for j in range(n_start_step_x + 1, n_end_step_x + 1):
            mask[i, j] = 0
    return mask, n_start_step_x, n_start_step_y, n_end_step_x, n_end_step_y

This function is used to exclude solid regions from the computations, ensuring that equations are only solved in fluid regions.

Oscillations and Check-Board Pattern with Vanishing Artificial Pressure

As the artificial pressure approaches zero, a check-board pattern emerges in the pressure domain due to the decoupling between the pressure and velocity fields. This pattern, indicative of instability, occurs because the artificial pressure term, which typically stabilizes the coupling between pressure and velocity, is no longer effective. The figure below illustrates this issue, where a stable pressure field is observed with non-zero artificial pressure, while a check-board pattern appears when artificial pressure is reduced to zero.

Results

The implementation of the artificial pressure method was validated using two benchmark problems: the cavity benchmark problem and the forward-step benchmark problem. These test cases were selected to evaluate the performance of the method in different flow configurations.

• Cavity Benchmark Problem: This problem involves fluid flow within a square cavity, where the bottem boundary is moving, and the other boundaries are stationary. It serves as a classical test for incompressible flow solvers, testing the method’s ability to handle flow patterns like recirculating regions and vortex formation.

• Forward-Step Benchmark Problem: In this test, the flow encounters a step in a channel, creating a separation bubble. This problem is used to assess how well the method captures the pressure and velocity fields around solid obstacles, testing the ability to resolve the pressure drop and the vortex structure formed downstream of the step.

The following results show the velocity and pressure fields obtained from these two benchmark cases, demonstrating the accuracy and stability of the solution under various flow conditions.