Adds a pertubation advection open boundary matching scheme #3977

jagoosw · 2024-12-06T19:01:09Z

This PR adds a PertubationAdvection open boundary condition which assumes that there is some mean flow (i.e. prescribed in a channel or from a coarser resolution model) which is homogeneous on the boundary but can vary in time.

On the boundary, we approximate (for an x-boundary) to:

$$\frac{\partial u}{\partial t} \approx -U\frac{\partial u\prime}{\partial x},$$

where $u=u\prime+U$
I have chosen to take an explicit/backwards Euler step:

$$\frac{u\prime(t^{n+1}) - u\prime(t^n)}{\Delta t} + \frac{U\prime(t^{n+1}) - U\prime(t^n)}{\Delta t} \approx -U\frac{u\prime(x_b, t^{n+1}) - u\prime(x_{b-1}, t^{n+1})}{\Delta x},$$

because then we don't have to store any other information (i.e. if we did a forward step we would need $u\prime(x_{b-1}, t^n)$ ). This results in:

$$u(t^{n+1}) = \frac{u^n+\tilde{U}u(x_{b-1}, t^{n+1})}{1+\tilde{U}},$$

where $\tilde{U}=\min\left(1, U\Delta t / \Delta x\right)$. I have also added restoring to $U$ (i.e. damping $u\prime\to0$) in some timescale $\tau$ which can be different for inflow and outflow to allow the scheme to work when there is inflow, which results in:

$$u(t^{n+1}) = \frac{u^n+\tilde{U}u(x_{b-1}, t^{n+1}) + U\tilde{\tau}}{1+\tilde{U}+\tilde{\tau}},$$

where $\tilde{\tau}=\Delta t / \tau$.

This has the limitation that if we want to extend to have wall tangent mean velocity ($\partial_t u \approx -(\vec{U}\cdot\nabla) u$) then we either have to solve the whole boundary at once since every point is going to depend on its boundary neighbours, or we need to take a forward Euler step in which case we need the first interior point at the previous time step.

I think we probably will need to address this for real nesting cases at some point.

u.mp4

For now, it seems to be working okay (left plot vorticity, right plot x-velocity):

(old plot)

https://github.com/user-attachments/assets/21ae3eb2-5d3b-4c08-86fb-bb9c222e8e6a

and the drag coefficient is ~1.4 at Re = 100 and I'm running a case at Re=1000 now.

I have also checked the Strouhal number which matches exactly to the expected values (~0.17).

jagoosw · 2024-12-06T19:03:32Z

@tomchor, do you think something like this:

Oceananigans.jl/validation/open_boundaries/cylinder.jl

Lines 18 to 76 in 32cd354

    
           """ 
        
               drag(modell; bounding_box = (-1, 3, -2, 2), ν = 1e-3) 
        
           Returns the drag within the `bounding_box` computed by: 
        
           ```math 
        
           \\frac{\\partial \\vec{u}}{\\partial t} + (\\vec{u}\\cdot\\nabla)\\vec{u}=-\\nabla P + \\nabla\\cdot\\vec{\\tau} + \\vec{F},\\newline 
        
           \\vec{F}_T=\\int_\\Omega\\vec{F}dV = \\int_\\Omega\\left(\\frac{\\partial \\vec{u}}{\\partial t} + (\\vec{u}\\cdot\\nabla)\\vec{u}+\\nabla P - \\nabla\\cdot\\vec{\\tau}\\right)dV,\\newline 
        
           \\vec{F}_T=\\int_\\Omega\\left(\\frac{\\partial \\vec{u}}{\\partial t}\\right)dV + \\oint_{\\partial\\Omega}\\left(\\vec{u}(\\vec{u}\\cdot\\hat{n}) + P\\hat{n} - \\vec{\\tau}\\cdot\\hat{n}\\right)dS,\\newline 
        
           \\F_u=\\int_\\Omega\\left(\\frac{\\partial u}{\\partial t}\\right)dV + \\oint_{\\partial\\Omega}\\left(u(\\vec{u}\\cdot\\hat{n}) - \\tau_{xx}\\right)dS + \\int_{\\partial\\Omega}P\\hat{x}\\cdot d\\vec{S},\\newline 
        
           F_u=\\int_\\Omega\\left(\\frac{\\partial u}{\\partial t}\\right)dV - \\int_{\\partial\\Omega_1} \\left(u^2 - 2\\nu\\frac{\\partial u}{\\partial x} + P\\right)dS + \\int_{\\partial\\Omega_2}\\left(u^2 - 2\\nu\\frac{\\partial u}{\\partial x}+P\\right)dS -  \\int_{\\partial\\Omega_2} uvdS + \\int_{\\partial\\Omega_4} uvdS, 
        
           ``` 
        
           where the bounding box is ``\\Omega`` which is formed from the boundaries ``\\partial\\Omega_{1}``, ``\\partial\\Omega_{2}``, ``\\partial\\Omega_{3}``, and ``\\partial\\Omega_{4}``  
        
           which have outward directed normals ``-\\hat{x}``, ``\\hat{x}``, ``-\\hat{y}``, and ``\\hat{y}`` 
        
           """ 
        
           function drag(model; 
        
                         bounding_box = (-1, 3, -2, 2), 
        
                         ν = 1e-3) 
        
               u, v, _ = model.velocities 
        
               uᶜ = Field(@at (Center, Center, Center) u) 
        
               vᶜ = Field(@at (Center, Center, Center) v) 
        
               xc, yc, _ = nodes(uᶜ) 
        
               i₁ = findfirst(xc .> bounding_box[1]) 
        
               i₂ = findlast(xc .< bounding_box[2]) 
        
               j₁ = findfirst(yc .> bounding_box[3]) 
        
               j₂ = findlast(yc .< bounding_box[4]) 
        
               uₗ² = Field(uᶜ^2, indices = (i₁, j₁:j₂, 1)) 
        
               uᵣ² = Field(uᶜ^2, indices = (i₂, j₁:j₂, 1)) 
        
               uvₗ = Field(uᶜ*vᶜ, indices = (i₁:i₂, j₁, 1)) 
        
               uvᵣ = Field(uᶜ*vᶜ, indices = (i₁:i₂, j₂, 1)) 
        
               ∂₁uₗ = Field(∂x(uᶜ), indices = (i₁, j₁:j₂, 1)) 
        
               ∂₁uᵣ = Field(∂x(uᶜ), indices = (i₂, j₁:j₂, 1)) 
        
               ∂ₜuᶜ = Field(@at (Center, Center, Center) model.timestepper.Gⁿ.u) 
        
               ∂ₜu = Field(∂ₜuᶜ, indices = (i₁:i₂, j₁:j₂, 1)) 
        
               p = model.pressures.pNHS 
        
               ∫∂ₓp = Field(∂x(p), indices = (i₁:i₂, j₁:j₂, 1)) 
        
               a_local = Field(Integral(∂ₜu)) 
        
               a_flux = Field(Integral(uᵣ²)) - Field(Integral(uₗ²)) + Field(Integral(uvᵣ)) - Field(Integral(uvₗ)) 
        
               a_viscous_stress = 2ν * (Field(Integral(∂₁uᵣ)) - Field(Integral(∂₁uₗ))) 
        
               a_pressure = Field(Integral(∫∂ₓp)) 
        
               return a_local + a_flux + a_pressure - a_viscous_stress 
        
           end

could belong in Oceanostics?

jagoosw · 2024-12-06T19:04:33Z

I have also run the same case with different domain lengths (6m, 12m - shown, 18m and 30m) and they all produce very similar Cd and St

tomchor · 2024-12-06T19:09:13Z

@tomchor, do you think something like this:

Oceananigans.jl/validation/open_boundaries/cylinder.jl

Lines 18 to 76 in 32cd354

"""

drag(modell; bounding_box = (-1, 3, -2, 2), ν = 1e-3)

Returns the drag within the `bounding_box` computed by:

```math

\\frac{\\partial \\vec{u}}{\\partial t} + (\\vec{u}\\cdot\\nabla)\\vec{u}=-\\nabla P + \\nabla\\cdot\\vec{\\tau} + \\vec{F},\\newline

\\vec{F}_T=\\int_\\Omega\\vec{F}dV = \\int_\\Omega\\left(\\frac{\\partial \\vec{u}}{\\partial t} + (\\vec{u}\\cdot\\nabla)\\vec{u}+\\nabla P - \\nabla\\cdot\\vec{\\tau}\\right)dV,\\newline

\\vec{F}_T=\\int_\\Omega\\left(\\frac{\\partial \\vec{u}}{\\partial t}\\right)dV + \\oint_{\\partial\\Omega}\\left(\\vec{u}(\\vec{u}\\cdot\\hat{n}) + P\\hat{n} - \\vec{\\tau}\\cdot\\hat{n}\\right)dS,\\newline

\\F_u=\\int_\\Omega\\left(\\frac{\\partial u}{\\partial t}\\right)dV + \\oint_{\\partial\\Omega}\\left(u(\\vec{u}\\cdot\\hat{n}) - \\tau_{xx}\\right)dS + \\int_{\\partial\\Omega}P\\hat{x}\\cdot d\\vec{S},\\newline

F_u=\\int_\\Omega\\left(\\frac{\\partial u}{\\partial t}\\right)dV - \\int_{\\partial\\Omega_1} \\left(u^2 - 2\\nu\\frac{\\partial u}{\\partial x} + P\\right)dS + \\int_{\\partial\\Omega_2}\\left(u^2 - 2\\nu\\frac{\\partial u}{\\partial x}+P\\right)dS - \\int_{\\partial\\Omega_2} uvdS + \\int_{\\partial\\Omega_4} uvdS,

```

where the bounding box is ``\\Omega`` which is formed from the boundaries ``\\partial\\Omega_{1}``, ``\\partial\\Omega_{2}``, ``\\partial\\Omega_{3}``, and ``\\partial\\Omega_{4}``

which have outward directed normals ``-\\hat{x}``, ``\\hat{x}``, ``-\\hat{y}``, and ``\\hat{y}``

"""

function drag(model;

bounding_box = (-1, 3, -2, 2),

ν = 1e-3)

u, v, _ = model.velocities

uᶜ = Field(@at (Center, Center, Center) u)

vᶜ = Field(@at (Center, Center, Center) v)

xc, yc, _ = nodes(uᶜ)

i₁ = findfirst(xc .> bounding_box[1])

i₂ = findlast(xc .< bounding_box[2])

j₁ = findfirst(yc .> bounding_box[3])

j₂ = findlast(yc .< bounding_box[4])

uₗ² = Field(uᶜ^2, indices = (i₁, j₁:j₂, 1))

uᵣ² = Field(uᶜ^2, indices = (i₂, j₁:j₂, 1))

uvₗ = Field(uᶜ*vᶜ, indices = (i₁:i₂, j₁, 1))

uvᵣ = Field(uᶜ*vᶜ, indices = (i₁:i₂, j₂, 1))

∂₁uₗ = Field(∂x(uᶜ), indices = (i₁, j₁:j₂, 1))

∂₁uᵣ = Field(∂x(uᶜ), indices = (i₂, j₁:j₂, 1))

∂ₜuᶜ = Field(@at (Center, Center, Center) model.timestepper.Gⁿ.u)

∂ₜu = Field(∂ₜuᶜ, indices = (i₁:i₂, j₁:j₂, 1))

p = model.pressures.pNHS

∫∂ₓp = Field(∂x(p), indices = (i₁:i₂, j₁:j₂, 1))

a_local = Field(Integral(∂ₜu))

a_flux = Field(Integral(uᵣ²)) - Field(Integral(uₗ²)) + Field(Integral(uvᵣ)) - Field(Integral(uvₗ))

a_viscous_stress = 2ν * (Field(Integral(∂₁uᵣ)) - Field(Integral(∂₁uₗ)))

a_pressure = Field(Integral(∫∂ₓp))

return a_local + a_flux + a_pressure - a_viscous_stress

end

could belong in Oceanostics?

Sure! Why not? :)

tomchor · 2024-12-06T19:14:35Z

src/BoundaryConditions/perturbation_advection_open_boundary_matching_scheme.jl

+@inline function _fill_east_halo!(j, k, grid, u, bc::PAOBC, loc::Tuple{Face, Any, Any}, clock, model_fields)
+    i = grid.Nx + 1
+
+    Δt = clock.last_stage_Δt
+
+    Δt = ifelse(isinf(Δt), 0, Δt)
+
+    Δx = xspacing(i, j, k, grid, loc...)
+
+    ūⁿ⁺¹ = getbc(bc, j, k, grid, clock, model_fields)
+
+    uᵢⁿ     = @inbounds u[i, j, k]
+    uᵢ₋₁ⁿ⁺¹ = @inbounds u[i - 1, j, k]
+
+    U = max(0, min(1, Δt / Δx * ūⁿ⁺¹))
+
+    τ = ifelse(ūⁿ⁺¹ >= 0, 
+               bc.classification.matching_scheme.outflow_timescale, 
+               bc.classification.matching_scheme.inflow_timescale)
+
+
+    τ̃ = Δt / τ
+
+    uᵢⁿ⁺¹ = (uᵢⁿ + U * uᵢ₋₁ⁿ⁺¹ + ūⁿ⁺¹ * τ̃) / (1 + τ̃ + U)
+
+    @inbounds u[i, j, k] = uᵢⁿ⁺¹#ifelse(active_cell(i, j, k, grid), uᵢⁿ⁺¹, zero(grid))
+end


I think we could implement everything using getindex() so that we implement it once (or at max twice; one for left and other for right BC) and can use it in three dimensions.

For example this

uᵢⁿ = @inbounds u[i, j, k] uᵢ₋₁ⁿ⁺¹ = @inbounds u[i - 1, j, k]

would become

uᵢⁿ = @inbounds getindex(u, boundary_indices...) uᵢ₋₁ⁿ⁺¹ = @inbounds getindex(u, one_off_boundary_indices...)

This would avoid errors when transcribing to other dimensions (I remember catching a couple for the flat extrapolation matching scheme).

yeah, I was trying to think of a clean way to do this

Attempted this now (but I've still not written out the other directions in case we want to change things first)

validation/open_boundaries/cylinder.jl

glwagner · 2024-12-06T20:45:15Z

There's a typo in your top equation right? I think it should be $\partial_t (u + U) = U \partial_x u$.

glwagner · 2024-12-06T20:46:30Z

A problem I think this scheme might have is that as the flow changes direction the assumption that the perturbation is small falls apart, which is maybe why it's been a bit unstable so far in an oscillating case.

What do you mean? What is the consequence of this approximation?

jagoosw · 2024-12-06T20:57:23Z

What do you mean? What is the consequence of this approximation?

I was thinking about this more and really to remain valid then in :

$$\partial_t (u\prime+U)\approx u\prime\partial_xu\prime + U\partial_xu\prime + (U - u\prime)/\tau$$

as $U\to0$ then we need $u\prime\partial_xu\prime\ll u\prime/\tau$ to still be able to throw away the first term, so I think its actually okay as long as you have the relaxation.

I found a typo in the west boundary which might be why the oscillating case wasn't working, rerunning now

jagoosw · 2024-12-06T20:57:31Z

There's a typo in your top equation right? I think it should be ∂ t ( u + U ) = U ∂ x u .
Yeah thanks for catching!

glwagner · 2024-12-06T21:57:59Z

What do you mean? What is the consequence of this approximation?

I was thinking about this more and really to remain valid then in :

as U → 0 then we need u ′ ∂ x u ′ ≪ u ′ / τ to still be able to throw away the first term, so I think its actually okay as long as you have the relaxation.

I found a typo in the west boundary which might be why the oscillating case wasn't working, rerunning now

Okay, I agree with you. You can also analyze the update formula / backward Euler step in the limit U -> 0. It doesn't look like it needs to be regularized in any way in that case though, it looks ok.

tomchor · 2024-12-09T19:14:17Z

Is there a reference for this method? If there is, please include it on the methods docstring. If not, it'd be good to include a brief explanation like the one at the top comment in the dosctring. I don't think this radiation method is very well known.

src/boundary_mean.jl

tomchor · 2025-01-21T03:01:00Z

test/test_boundary_conditions_integration.jl

+
+        u = normal_velocity(Val(orientation), model)
+
+        CUDA.@allowscalar u.data .= 1


I think we're trying to avoid introducing new @alloscalar instances to testing. @glwagner can you confirm?

I thought the problem was that tests currently have a global @allowscalar, but some tests need them or will be more complicated to write?

I think you can write most of them simply without allowscalar, but I'm not an expert on this. For example you may be able to write something like interior(u) .= 1 without it (I haven't tested it though), and it's a pretty simple change.

I say try to avoid it with simple statements like the one I just mentioned. If we can't do it simply and @glwagner doesn't manifest any opinions on it, we move forward. How does that sound?

I sorted this now, it wasn't as straight forward as just using interior because I needed to set a peripheral node, but I just used view instead

test/test_boundary_conditions_integration.jl

validation/open_boundaries/cylinder.jl

tomchor

Made a few comments, but it looks like there are old unresolved comments and some files here and there which do not seem to be ready for prime yet. let me know if anything needs clarification.

Co-authored-by: Tomas Chor <[email protected]>

jagoosw · 2025-01-21T15:40:35Z

@tomchor I think I've addressed your comments now

validation/open_boundaries/cylinder.jl

tomchor · 2025-01-21T15:49:08Z

validation/open_boundaries/cylinder.jl


-ax = Axis(fig[1, 1], aspect = DataAspect())
+    simulation.output_writers[:drag] = JLD2OutputWriter(model, (; drag_force),
+                                                        schedule = IterationInterval(Int(0.1/Δt)),#TimeInterval(0.1),


Suggested change

schedule = IterationInterval(Int(0.1/Δt)),#TimeInterval(0.1),

schedule = IterationInterval(Int(0.1/Δt)),

tomchor · 2025-01-21T15:49:23Z

validation/open_boundaries/cylinder.jl

+    outputs = (; u, v, p, ζ)
+
+    simulation.output_writers[:jld2] = JLD2OutputWriter(model, outputs,
+                                                        schedule = IterationInterval(Int(2/Δt)),#TimeInterval(0.1),


Suggested change

schedule = IterationInterval(Int(2/Δt)),#TimeInterval(0.1),

schedule = IterationInterval(Int(2/Δt)),

tomchor · 2025-01-21T15:49:38Z

validation/open_boundaries/cylinder.jl


-progress(sim) = @info "$(time(sim)) with Δt = $(prettytime(sim.Δt)) in $(prettytime(sim.run_wall_time))"
+    simulation = Simulation(model; Δt, stop_time)
+    #conjure_time_step_wizard!(simulation, cfl=1.0, IterationInterval(3); max_Δt)


Suggested change

#conjure_time_step_wizard!(simulation, cfl=1.0, IterationInterval(3); max_Δt)

tomchor · 2025-01-21T15:56:00Z

validation/open_boundaries/cylinder.jl

+# there is some problem using ConjugateGradientPoissonSolver with TimeInterval because the timestep can go really small
+# so while I identify the issue I'm using IterationInterval and a fixed timestep


Have you tried setting minimum_relative_step? (see #3606)

This seems to fix the issue! I hadn't tried this since early november/december so I hadn't seen you'd made this fix so this is great, thank you

I'll update the validation script

Co-authored-by: Tomas Chor <[email protected]>

validation/open_boundaries/cylinder.jl

jagoosw · 2025-01-27T19:26:57Z

src/BoundaryConditions/perturbation_advection_open_boundary_matching_scheme.jl

+We begin with the equation governing the fluid in the interior:
+    ∂ₜu⃗ + u⃗⋅∇u⃗ = −∇P + F⃗,
+and note that on the boundary the pressure gradient is zero.
+We can then assume that the flow composes of mean (U⃗) and pertubation (u⃗′) components,
+and considering the x-component of velocity, we can rewrite the equation as
+    ∂ₜu₁ = -u₁∂₁u - u₂∂₂u₁ - u₃∂₃u₁ + F₁ ≈ - U₁∂₁u₁′ - U₂∂₂u₁′ - U₃∂₃u₁′ + F.
+
+Simplify by assuming that U⃗ = Ux̂, an then take a numerical step to find u₁.
+
+When the boundaries are filled the interior is at time tₙ₊₁ so we can take
+a backwards euler step (in the case that the mean flow is boundary normal) on a right boundary:
+    (Uⁿ⁺¹ - Uⁿ) / Δt + (u′ⁿ⁺¹ - u′ⁿ) / Δt = - Uⁿ⁺¹ (u′ⁿ⁺¹ᵢ - u′ⁿ⁺¹ᵢ₋₁) / Δx + Fᵤ.
+
+This can not be solved for general forcing, but if we assume the dominant forcing is
+relaxation to the mean velocity (i.e. u′→0) then Fᵤ = -u′ / τ then we can find u′ⁿ⁺¹:
+    u′ⁿ⁺¹ = (uⁿ + Ũu′ⁿ⁺¹ᵢ₋₁ - Uⁿ⁺¹) / (1 + Ũ + Δt/τ),
+
+where Ũ = U Δt / Δx, then uⁿ⁺¹ is:
+    uⁿ⁺¹ = (uᵢⁿ + Ũuᵢ₋₁ⁿ⁺¹ + Uⁿ⁺¹τ̃) / (1 + τ̃ + U)
+
+where τ̃ = Δt/τ.


Is this not rendered correctly too @tomchor ?

If I use FiraCode on chrome then some of the stuff in the first line of code doesn't render correctly:

On firefox it all renders fine:

And when I manually switch to JuliaMono it all renders fine.

But after finding out that this is all browser/machine dependent, I do think we shouldn't spend too much time trying to "fix" this. I say we move on from tweaking the unicode and address the rest of the comments so that we can merge.

jagoosw and others added 3 commits November 18, 2024 17:32

fix FEOBC

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Unexpected FlatExtrapolationBoundaryCondition behaviour in validation script cylinder.jl #3975

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jagoosw changed the title ~~[WIP] Adds a pertubation advection open boundary matching scheme~~ Adds a pertubation advection open boundary matching scheme Jan 22, 2025

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Immediate NaNs running a non-hydrostatic model with an immersed boundary and open boundary conditions and immersed pressure solver #3831

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jagoosw Jan 21, 2025

jagoosw Jan 21, 2025

jagoosw Jan 27, 2025

tomchor Jan 27, 2025


		u = normal_velocity(Val(orientation), model)

		CUDA.@allowscalar u.data .= 1

	schedule = IterationInterval(Int(0.1/Δt)),#TimeInterval(0.1),
	schedule = IterationInterval(Int(0.1/Δt)),

	schedule = IterationInterval(Int(2/Δt)),#TimeInterval(0.1),
	schedule = IterationInterval(Int(2/Δt)),

		# there is some problem using ConjugateGradientPoissonSolver with TimeInterval because the timestep can go really small
		# so while I identify the issue I'm using IterationInterval and a fixed timestep

Adds a pertubation advection open boundary matching scheme #3977

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Adds a pertubation advection open boundary matching scheme #3977

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jagoosw commented Dec 6, 2024 • edited Loading

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jagoosw commented Dec 6, 2024 •

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glwagner commented Dec 6, 2024 •

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glwagner commented Dec 6, 2024 •

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