Solve a Nonlinear Model of Fluid Flow

Model the time-dependent flow around a cylinder with the transient Navier–Stokes equation

Consider the system of differential equations modeling the time-dependent flow around a cylinder with the transient Navier–Stokes equation. More details about this model can be found in the documentation. Here

is the vector-valued velocity field,

is the pressure and

the identity matrix.

and

are the density and viscosity, respectively.

∂u ∂t +∇(μ∇u)+ρu∇u+Ip	=	0
∇u	=	0.

Set up parameters and a region:

In[1]:=

rules={length22/10,height41/100};Ω=RegionDifference[Rectangle[{0,0},{length,height}],Disk[{1/5,1/5},1/20]]/.rules;

Visualize the region and the inflow profile. The arrows represent the incoming flow, which is ramped up over time:

In[2]:=

ShowBoundaryDiscretizeRegion[Ω],bmesh=HighlightMeshBoundaryDiscretizeRegionΩ,

...

,

...

,VectorPlot{4*1.5*y*(0.41-y)/0.41^2,0},{x,0,2.2},{y,0,0.41},

,ImageSizeMedium

Out[2]=

Specify the transient viscous Navier–Stokes equation with viscosity

and density

In[3]:=

op=ρ

(1,0,0)

[t,x,y]+ρ{u[t,x,y],v[t,x,y]}.

∇

{x,y}

u[t,x,y]+

∇

{x,y}

·-μ

∇

{x,y}

u[t,x,y]+

(0,1,0)

[t,x,y],ρ

(1,0,0)

[t,x,y]+ρ{u[t,x,y],v[t,x,y]}.

∇

{x,y}

v[t,x,y]+

∇

{x,y}

·-μ

∇

{x,y}

v[t,x,y]+

(0,0,1)

[t,x,y],

(0,1,0)

[t,x,y]+

(0,0,1)

[t,x,y]/.{μ10^-3,ρ1};

Specify a function that ramps up the inflow velocity:

In[4]:=

ramp=Function[t,Exp[5*t]/(Exp[20]+Exp[5*t])];

Specify the boundary conditions for inflow on the left and right, the no-slip boundary conditions on the remaining walls: inflow boundary condition on the left:

In[5]:=

inflowBC=DirichletCondition[{u[t,x,y]ramp[t]*4*1.5*y*(height-y)/height^2,v[t,x,y]0},x0];outflowBC=DirichletCondition[p[t,x,y]0.,xlength];wallBC=DirichletCondition[{u[t,x,y]0,v[t,x,y]0},0<x<length];

Obtain a list of all boundary conditions and replace the parameters with the chosen values:

In[6]:=

bcs={inflowBC,outflowBC,wallBC}/.rules;

Set up initial conditions such that the system is at rest:

In[7]:=

ic={u[0,x,y]0,v[0,x,y]0,p[0,x,y]0};

Time-integrate the Navier–Stokes equation on a mesh with specified spacing while interpolating the velocities with second order and the pressure with first order. This calculation can take a few minutes, but the progress is reported with a

Monitor

In[8]:=

Monitor[AbsoluteTiming[{xVel,yVel,pressure}=NDSolveValue[{op{0,0,0},bcs,ic},{u,v,p},{x,y}∈Ω,{t,0,10},Method{"PDEDiscretization"{"MethodOfLines","SpatialDiscretization"{"FiniteElement","InterpolationOrder"{u2,v2,p1},"MeshOptions"{"MaxCellMeasure"0.0005}}}},EvaluationMonitor(currentTime=Row[{"t = ",ToString[t]}])];],currentTime]

Out[8]=

{124.154,Null}

Compute the vorticity of the flow field:

In[9]:=

vorticity=Function@@List[{t,x,y},-D[xVel[t,x,y],y]+D[yVel[t,x,y],x]];

Set up the color scheme to be used for the visualization:

In[10]:=

cc=Join[{-30},N@Subdivide[-20,20,50],{30}];colors=ColorData["TemperatureMap"][Rescale[#,MinMax[cc]]]&/@cc;

Visualize the vorticity after 8 time units have passed:

In[11]:=

ShowContourPlotvorticity[8,x,y],{x,0,2.2},{y,0,0.41},

,



Out[11]=

Generate the frames for an animation:

In[12]:=

frames=TableShowbmesh,ContourPlotvorticity[t,x,y],{x,0,2.2},{y,0,0.41},

,{t,3,10,1/6};

In[13]:=

FrameListVideo[frames,FrameRate5]

Out[13]=

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Fluid Flow

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Contributed by: Wolfram Staff

Solve a Nonlinear Model of Fluid Flow

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