Grid operations
Basic differential operations
There are a variety of (purely) grid-based operators that are useful for carrying out calculations in immersed layer problems. We will demonstrate a few of them here. We will start by generating the cache, just as we did in Immersed layer caches
using ImmersedLayers
using PlotsSet up a grid and cache
Δx = 0.01
Lx = 4.0
xlim = (-Lx/2,Lx/2)
ylim = (-Lx/2,Lx/2)
g = PhysicalGrid(xlim,ylim,Δx)PhysicalGrid{2}((404, 404), (202, 202), 0.01, ((-2.0100000000000002, 2.0100000000000002), (-2.0100000000000002, 2.0100000000000002)), 4)We still generate a cache for these operations, but now, we only supply the grid. There are no immersed surfaces for this demonstration.
cache = SurfaceScalarCache(g,scaling=GridScaling)Surface cache with scaling of type GridScaling
0 point data of type ScalarData{0, Float64, Vector{Float64}}
Grid data of type Nodes{Primal, 404, 404, Float64, Matrix{Float64}}
To demonstrate, let's generate a Gaussian
p = zeros_grid(cache)
xg, yg = x_grid(cache), y_grid(cache)
p .= exp.(-(xg∘xg)-(yg∘yg))Nodes{Primal, 404, 404, Float64, Matrix{Float64}} data
Printing in grid orientation (lower left is (1,1))
403×403 Matrix{Float64}:
0.000309609 0.000322277 0.000335396 … 0.000322277 0.000309609
0.000322277 0.000335463 0.000349118 0.000335463 0.000322277
0.000335396 0.000349118 0.00036333 0.000349118 0.000335396
0.000348979 0.000363257 0.000378044 0.000363257 0.000348979
0.000363039 0.000377893 0.000393276 0.000377893 0.000363039
0.000377591 0.00039304 0.000409039 … 0.00039304 0.000377591
0.000392647 0.000408712 0.000425349 0.000408712 0.000392647
0.000408222 0.000424924 0.000442221 0.000424924 0.000408222
0.00042433 0.000441691 0.000459671 0.000441691 0.00042433
0.000440985 0.000459028 0.000477713 0.000459028 0.000440985
⋮ ⋱
0.00042433 0.000441691 0.000459671 0.000441691 0.00042433
0.000408222 0.000424924 0.000442221 … 0.000424924 0.000408222
0.000392647 0.000408712 0.000425349 0.000408712 0.000392647
0.000377591 0.00039304 0.000409039 0.00039304 0.000377591
0.000363039 0.000377893 0.000393276 0.000377893 0.000363039
0.000348979 0.000363257 0.000378044 0.000363257 0.000348979
0.000335396 0.000349118 0.00036333 … 0.000349118 0.000335396
0.000322277 0.000335463 0.000349118 0.000335463 0.000322277
0.000309609 0.000322277 0.000335396 0.000322277 0.000309609Now, let's generate the gradient of these data
v = zeros_gridgrad(cache)
grad!(v,p,cache)
plot(v,cache)
We can then compute the derivative of this data
divv = zeros_grid(cache)
divergence!(divv,v,cache)
plot(divv,cache)
Convective derivatives
And finally, let's compute convective derivatives. First, we will compute
\[\mathbf{v}\cdot\nabla p\]
For this operation, we create a special additional cache using ConvectiveDerivativeCache. This extra cache holds additional memory for making the calculation of the convective derivative faster if we compute it often.
cdcache = ConvectiveDerivativeCache(cache)
vdp = zeros_grid(cache)
@time convective_derivative!(vdp,v,p,cache,cdcache) 0.001214 seconds (18 allocations: 3.078 KiB)Plot it
plot(vdp,cache)
Now, let's compute
\[\mathbf{v}\cdot\nabla\mathbf{v}\]
For this, we create a cache for VectorGridData, and a new instance of ConvectiveDerivativeCache to go along with it.
vcache = SurfaceVectorCache(g,scaling=GridScaling)
vdv = zeros_grid(vcache)
cdvcache = ConvectiveDerivativeCache(vcache)
@time convective_derivative!(vdv,v,vcache,cdvcache) 0.003130 seconds (55 allocations: 8.875 KiB)Plot it
plot(vdv,vcache)
Finally, let's compute
\[(\curl\mathbf{v})\times\mathbf{v}\]
For this, we create a cache called RotConvectiveDerivativeCache to go along with it.
w = zeros_gridcurl(vcache)
curl!(w,v,vcache)
wv = zeros_grid(vcache)
cdrcache = RotConvectiveDerivativeCache(vcache)
@time w_cross_v!(wv,w,v,vcache,cdrcache)Edges{Primal, 404, 404, Float64, Vector{Float64}} data
u (in grid orientation)
403×404 Matrix{Float64}:
0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋮ ⋱ ⋮
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 … -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 … -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
v (in grid orientation)
404×403 Matrix{Float64}:
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
⋮ ⋮ ⋱ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0Plot it
plot(wv,vcache)
Grid operator functions
CartesianGrids.divergence! — Functiondivergence!(p::Nodes{Primal/Dual},v::Edges{Primal/Dual},cache::BasicILMCache)
divergence!(p::Nodes{Primal/Dual},v::Edges{Primal/Dual},sys::ILMSystem)Compute the discrete divergence of v and return it in p, scaling it by the grid spacing if cache (or sys) is of GridScaling type, or leaving it as a simple differencing if cache (or sys) is of IndexScaling type.
CartesianGrids.grad! — Functiongrad!(v::Edges{Primal/Dual},p::Nodes{Primal/Dual},cache::BasicILMCache)
grad!(v::Edges{Primal/Dual},p::Nodes{Primal/Dual},sys::ILMSystem)Compute the discrete gradient of p and return it in v, scaling it by the grid spacing if cache (or sys) is of GridScaling type, or leaving it as a simple differencing if cache (or sys) is of IndexScaling type.
grad!(v::EdgeGradient{Primal/Dual,Dual/Primal},p::Edges{Primal/Dual},cache::BasicILMCache)
grad!(v::EdgeGradient{Primal/Dual,Dual/Primal},p::Edges{Primal/Dual},sys::ILMSystem)Compute the discrete gradient of edge data p and return it in v, scaling it by the grid spacing if cache (or sys) is of GridScaling type, or leaving it as a simple differencing if cache (or sys) is of IndexScaling type.
CartesianGrids.curl! — Functioncurl!(v::Edges{Primal},s::Nodes{Dual},cache::BasicILMCache)
curl!(v::Edges{Primal},s::Nodes{Dual},sys::ILMSystem)Compute the discrete curl of s and return it in v, scaling it by the grid spacing if cache (or sys) is of GridScaling type, or leaving it as a simple differencing if cache (or sys) is of IndexScaling type.
curl!(w::Nodes{Dual},v::Edges{Primal},cache::BasicILMCache)
curl!(w::Nodes{Dual},v::Edges{Primal},sys::ILMSystem)Compute the discrete curl of v and return it in w, scaling it by the grid spacing if cache (or sys) is of GridScaling type, or leaving it as a simple differencing if cache (or sys) is of IndexScaling type.
CartesianGrids.convective_derivative! — Functionconvective_derivative!(vdp::Nodes{Primal},v::Edges,p::Nodes{Primal},base_cache::BasicILMCache,extra_cache::ConvectiveDerivativeCache)Compute the convective derivative of p with velocity v, i.e., $v\cdot \nabla p$, and return the result in vdp. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling. This version of the method uses extra_cache of type ConvectiveDerivativeCache.
convective_derivative!(vdw::Nodes{Dual},v::Edges,w::Nodes{Dual},base_cache::BasicILMCache,extra_cache::ConvectiveDerivativeCache)Compute the convective derivative of w with velocity v, i.e., $v\cdot \nabla w$, and return the result in vdw. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling. This version of the method uses extra_cache of type ConvectiveDerivativeCache.
convective_derivative!(vdu::Edges,v::Edges,u::Edges,base_cache::BasicILMCache,extra_cache::ConvectiveDerivativeCache)Compute the convective derivative of u, i.e., $v\cdot \nabla u$, and return the result in vdu. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling. This version of the method uses extra_cache of type ConvectiveDerivativeCache.
convective_derivative!(vdv::Edges,v::Edges,base_cache::BasicILMCache,extra_cache::ConvectiveDerivativeCache)Compute the convective derivative of v, i.e., $v\cdot \nabla v$, and return the result in vdv. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling. This version of the method uses extra_cache of type ConvectiveDerivativeCache.
ImmersedLayers.convective_derivative — Functionconvective_derivative(v::Edges{Primal},p::Nodes{Primal},base_cache::BasicILMCache)Compute the convective derivative of p with velocity v, i.e., $v\cdot \nabla p$. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling.
convective_derivative(v::Edges{Dual},w::Nodes{Dual},base_cache::BasicILMCache)Compute the convective derivative of w with velocity v, i.e., $v\cdot \nabla w$. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling.
convective_derivative(v::Edges,base_cache::BasicILMCache)Compute the convective derivative of v, i.e., $v\cdot \nabla v$. The result is either divided by unity or the grid cell size depending on whether base_cache is of type IndexScaling or GridScaling.
ImmersedLayers.ConvectiveDerivativeCache — TypeConvectiveDerivativeCache(dv::EdgeGradient)Create a cache (a subtype of AbstractExtraILMCache) for computing the convective derivative, using dv to define the cache data.
ImmersedLayers.w_cross_v! — Functionw_cross_v!(vw::Edges{Primal},w::Nodes{Dual},v::Edges{Primal},base_cache::BasicILMCache,extra_cache::RotConvectiveDerivativeCache)Compute the term w \times v, with vorticity w and velocity v, and return the result in vw. This version of the method uses extra_cache of type RotConvectiveDerivativeCache.
ImmersedLayers.w_cross_v — Functionw_cross_v(w::Nodes{Dual},v::Edges{Primal},base_cache::BasicILMCache)Compute the term w \times v, with vorticity w and velocity v.
ImmersedLayers.RotConvectiveDerivativeCache — TypeRotConvectiveDerivativeCache(w::Nodes{Dual})Create a cache (a subtype of AbstractExtraILMCache) for computing the convective derivative in rotational form, using w to define the cache data.
ImmersedLayers.laplacian! — Functionlaplacian!(w::GridData,s::GridData,sys::ILMSystem)Compute the Laplacian of grid data s, and divide the result by unity or by the grid cell size, depending on whether sys has IndexScaling or GridScaling, respectively, and return the result as w.
laplacian!(w::GridData,s::GridData,cache::BasicILMCache)Compute the Laplacian of grid data s, and divide the result by unity or by the grid cell size, depending on whether cache has IndexScaling or GridScaling, respectively, and return the result as w.
ImmersedLayers.inverse_laplacian! — Functioninverse_laplacian!(w::GridData,sys::ILMSystem)Compute the in-place inverse Laplacian of grid data w, and multiply the result by unity or by the grid cell size, depending on whether sys has IndexScaling or GridScaling, respectively.
inverse_laplacian!(w::GridData,cache::BasicILMCache)Compute the in-place inverse Laplacian of grid data w, and multiply the result by unity or by the grid cell size, depending on whether cache has IndexScaling or GridScaling, respectively.
inverse_laplacian!(sol::GridData,rhs::GridData,cache::BasicILMCache)Compute the iinverse Laplacian of grid data rhs, multiply the result by unity or by the grid cell size, depending on whether cache has IndexScaling or GridScaling, respectively, and return the result as sol.
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