EnM E 320 1D Navier-Stokes Problem For EnME 303

Enme 320 1 D Navier Stokes Problem For Enme 303one Dimensional Flow E

Enme 320 1-D Navier-Stokes problem for ENME 303. One-dimensional flow equation. Numerically solve the partial differential equation for one-dimensional flow between in a gap, also known as the heat equation: ∂u/∂t = ν ∂²u/∂x². Use a uniform grid across a channel with a width of one unit. The velocity profiles for Couette and Poiseuille (channel flow) between two flat plates at steady-state can be obtained by solving the algebraic equations resulting from discretizing the PDE using either an explicit or an implicit time-stepping method. The general form of the discretized PDEs is given by:

For an explicit time step:

[(uₙⱼ₊₁ - uₙⱼ)/Δt] = ν [(uₙⱼ₊₁ - 2uₙⱼ + uₙⱼ₋₁)/Δx²]

And for an implicit time step:

[(uₙ₊₁ⱼ - uₙⱼ)/Δt] = ν [(uₙ₊₁ⱼ₊₁ - 2uₙ₊₁ⱼ + uₙ₊₁ⱼ₋₁)/Δx²]

where y = (j - 1) Δy, with Δy = (channel width)/(N - 1), and N is the number of nodes. By rearranging the implicit scheme so that unknown velocity values at time n+1 are on the left side, the discretized equation becomes:

[(uₙ₊₁ⱼ) - uₙⱼ]/Δt = ν [ (uₙ₊₁ⱼ₊₁ - 2uₙ₊₁ⱼ + uₙ₊₁ⱼ₋₁)/Δx² ]

This leads to a system of algebraic equations for all j from 2 to N, requiring boundary conditions at the walls. The problem asks what initial velocity profile to assume. For Couette flow, set μ=1, dp/dx=0, with the lower wall velocity at u=1 and upper wall at u=0; the channel width is unity. Discretize with 11 and 21 nodes and experiment with your explicit code using different time steps (t=0.01, 0.1, 1, etc.). Observe how the solution converges to steady state and examine stability issues when the time step is too large. Then, implement an implicit scheme using a large time step (t=10^5) to find steady-state flow, calculate mass flux, and shear stress on the walls.

Paper For Above instruction

The one-dimensional Navier-Stokes equation, simplified as the heat/diffusion equation, provides a fundamental model for studying viscous flow behaviors between two parallel plates. This problem involves numerically solving the PDE using finite difference methods, with particular focus on understanding explicit versus implicit time-stepping schemes, their stability, and convergence properties. The approach leverages discretization of the channel into uniform grid points and applies boundary conditions representative of Couette flow, a classic shear-driven flow scenario.

To simulate Couette flow in this model, the boundary conditions are set such that the lower wall moves with velocity u=1, while the upper wall remains stationary with u=0. These conditions mimic a shear-driven flow where the fluid velocity transitions smoothly from the moving wall to the stationary one, establishing a velocity profile that at steady state becomes linear—a hallmark of viscous dominated flow without pressure gradient influences. The initial velocity profile can be uniform or set to zero throughout the domain, with the system evolving over time steps until the steady state is achieved.

The discretized form of the explicit and implicit schemes involves defining the nodal velocities at each grid point and updating them iteratively based on the previous time step. The explicit scheme calculates the new velocities directly from the known previous values, which imposes a strict limit on the maximum permissible time step for stability, known as the Courant-Friedrichs-Lewy (CFL) condition. In contrast, the implicit method, particularly using methods such as the Crank-Nicolson or backward Euler, allows for larger time steps due to its unconditionally stable nature, although it requires solving a system of algebraic equations at each iteration.

Experiments with different time steps reveal that smaller steps speed up convergence but increase computational time, whereas larger steps risk numerical instability or unphysical oscillations if stability conditions are violated. For explicit schemes, selecting t below the critical threshold results in smooth convergence to the steady state profile, which becomes linear as expected. Implicit schemes with large time steps demonstrate faster convergence in terms of iterations but demand the solution of matrix systems, reflecting a trade-off between accuracy and computational efficiency.

Furthermore, when applying a pressure gradient to induce channel flow, the velocity profile shifts from purely shear-driven to pressure-driven flow, resulting in a parabolic velocity distribution characteristic of laminar Poiseuille flow. Calculations of the mass flux and wall shear stresses from the numerical solutions provide additional insights into the flow dynamics. The mass flux, derived from integrating the velocity profile, indicates the volume of fluid passing through the channel per unit time, whereas the shear stress reflects the viscous forces acting on the channel walls, critical for engineering applications involving lubrication, blood flow, and microfluidics.

In conclusion, solving the one-dimensional Navier-Stokes or heat equation numerically using finite difference methods offers valuable insights into fluid flow phenomena. Understanding the stability constraints, convergence rates, and physical implications of boundary conditions enhances the accuracy and efficiency of simulations. The ability to adapt time-stepping schemes based on problem parameters is essential for modeling practical flows in engineering systems accurately and efficiently.

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