Couette Flow

Couette flow is the steady viscous flow between two parallel plates where one plate is stationary and the other moves at a constant velocity. The exact solution is a linear velocity profile: $v_x(y) = U_w y / H$.

This tutorial demonstrates the NavierStokesEquations solver with NoSlipBC (stationary wall) and DirichletHyperbolicBC (moving wall).

Problem Setup

We solve in a channel $[0, 1] \times [0, H]$ with $H = 1$, periodic in $x$. The bottom wall is stationary and the top wall moves at velocity $U_w = 0.01$.

using FiniteVolumeMethod
using StaticArrays

gamma = 1.4
eos = IdealGasEOS(gamma)
mu = 0.01
Pr = 0.72
ns = NavierStokesEquations{2}(eos, mu = mu, Pr = Pr)

# Physical parameters (low Mach number)
rho0 = 1.0
P0 = 100.0
U_wall = 0.01
H = 1.0

1.0

We initialise with the exact Couette profile so the simulation remains near steady state:

w_top = SVector(rho0, U_wall, 0.0, P0)

function ic_couette(x, y)
    vx = U_wall * y / H
    return SVector(rho0, vx, 0.0, P0)
end

ic_couette (generic function with 1 method)

Boundary Conditions

• Left/Right: periodic (the flow is uniform in $x$)
• Bottom: NoSlipBC — negates all velocity components
• Top: DirichletHyperbolicBC — prescribes the moving wall state

nx, ny = 4, 16
mesh = StructuredMesh2D(0.0, 1.0, 0.0, H, nx, ny)

prob = HyperbolicProblem2D(
    ns, mesh, HLLCSolver(), NoReconstruction(),
    PeriodicHyperbolicBC(), PeriodicHyperbolicBC(),
    NoSlipBC(), DirichletHyperbolicBC(w_top),
    ic_couette; final_time = 0.5, cfl = 0.3
)
coords, U, t_final = solve_hyperbolic(prob)

4×16 Matrix{Tuple{Float64, Float64}}:
 (0.125, 0.03125)  (0.125, 0.09375)  …  (0.125, 0.90625)  (0.125, 0.96875)
 (0.375, 0.03125)  (0.375, 0.09375)     (0.375, 0.90625)  (0.375, 0.96875)
 (0.625, 0.03125)  (0.625, 0.09375)     (0.625, 0.90625)  (0.625, 0.96875)
 (0.875, 0.03125)  (0.875, 0.09375)     (0.875, 0.90625)  (0.875, 0.96875)

Comparison with Exact Solution

yc = [coords[1, j][2] for j in 1:ny]
# Take a slice at the middle x-cell
ix_mid = nx ÷ 2 + 1
vx_num = [conserved_to_primitive(ns, U[ix_mid, j])[2] for j in 1:ny]
vx_exact = [U_wall * y / H for y in yc]

max_err = maximum(abs.(vx_num .- vx_exact))

true

Visualisation

using CairoMakie

fig = Figure(fontsize = 24, size = (600, 500))
ax = Axis(
    fig[1, 1], xlabel = L"v_x", ylabel = "y",
    title = "Couette Flow: velocity profile"
)
lines!(ax, vx_exact, yc, color = :black, linewidth = 2, label = "Exact")
scatter!(ax, vx_num, yc, color = :blue, markersize = 10, label = "Numerical")
axislegend(ax, position = :lt)
resize_to_layout!(fig)
fig

The numerical solution closely matches the exact linear Couette profile. The maximum error is $(round(max_err, sigdigits=3)), which is small compared to the wall velocity $U_w = $(U_wall)$.

Just the code

An uncommented version of this example is given below. You can view the source code for this file here.

using FiniteVolumeMethod
using StaticArrays

gamma = 1.4
eos = IdealGasEOS(gamma)
mu = 0.01
Pr = 0.72
ns = NavierStokesEquations{2}(eos, mu = mu, Pr = Pr)

# Physical parameters (low Mach number)
rho0 = 1.0
P0 = 100.0
U_wall = 0.01
H = 1.0

w_top = SVector(rho0, U_wall, 0.0, P0)

function ic_couette(x, y)
    vx = U_wall * y / H
    return SVector(rho0, vx, 0.0, P0)
end

nx, ny = 4, 16
mesh = StructuredMesh2D(0.0, 1.0, 0.0, H, nx, ny)

prob = HyperbolicProblem2D(
    ns, mesh, HLLCSolver(), NoReconstruction(),
    PeriodicHyperbolicBC(), PeriodicHyperbolicBC(),
    NoSlipBC(), DirichletHyperbolicBC(w_top),
    ic_couette; final_time = 0.5, cfl = 0.3
)
coords, U, t_final = solve_hyperbolic(prob)

yc = [coords[1, j][2] for j in 1:ny]
# Take a slice at the middle x-cell
ix_mid = nx ÷ 2 + 1
vx_num = [conserved_to_primitive(ns, U[ix_mid, j])[2] for j in 1:ny]
vx_exact = [U_wall * y / H for y in yc]

max_err = maximum(abs.(vx_num .- vx_exact))

using CairoMakie

fig = Figure(fontsize = 24, size = (600, 500))
ax = Axis(
    fig[1, 1], xlabel = L"v_x", ylabel = "y",
    title = "Couette Flow: velocity profile"
)
lines!(ax, vx_exact, yc, color = :black, linewidth = 2, label = "Exact")
scatter!(ax, vx_num, yc, color = :blue, markersize = 10, label = "Numerical")
axislegend(ax, position = :lt)
resize_to_layout!(fig)
fig

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