1. Advection – Diffusion Equation
Lecture 16
Advection – Diffusion Equation

2. Objectives for Part II of the course
Part II aims to present resolution methods for the Advection-Diffusion Equation.
The problem will be ilustrated in a 1D system using the case of a river.
Four cases will be considered:
Transport of a conservative property (advection/diffusion only)
Transport with decay
Heat Transport
Transport of properties with internal reactions.
A VBA code will be used.

3. Numerics to be addressed
Reassessment of the finite-volume approach to quantify the conservation principle “The rate of accumulation inside a control volume is equal to the entrance minus the leaving rates, plus production minus consumption”.
Assessment of the numerical difficulties of advection. Upwind, central differences and “Quick” methods for advection. Numerical diffusion.
Time discretization and stability: Explicit, implicit and semi-implicit methods (Crank-Nicholson). Courant and Diffusion numbers.
Deduction of the algebraic equations from differential equations, using Taylor series. Accuracy and truncation error.

4. The Advection – Diffusion Equation
Advection is the transport by the Velocity
Diffusion is the transport due to random movements of matter not accounted by the velocity.
The advection – diffusion equation is derived from the conservation principle:
{The rate of accumulation of a property inside a control volume} =
{what flows in minus what flows out}+{Production minus Consumption}
Because it is derived from a conservation principle is often called Conservation Equation.
Because it describes temporal is also called Evolution Equation.
Because advective and diffusive transport are often important is also called Transport Equation.

5. What is velocity? For a body is clear
What would happen if car parts were not attached to each other?
Could we define the velocity of the car?

6. Velocity of a cyclist and velocity of the squad
Velocity of a Fluid molecule and velocity of the flow!!!
The velocity of the squad could be computed as the average of the cyclists’ velocities.
The velocity of the fluid could be computed as the average of the molecules’ velocities.
But we do not know them!!!
But we can know the flux, thus we can define velocity as:

7. Velocity and Advective flux
If velocity is not null, there is net flux across a surface. The Velocity is the volume crossing the surface, per unity of surface and per unit of time.
The value of the velocity obtained depends on the size of the surface used to measure it. In fact we measure the average velocity across that surface!!!
And if the surface is infinitesimal?
According to the continuum medium hypothesis it is still much larger then the distance between molecules and thus that velocity cannot consider the molecular Brownian movement.
Brownian transport must be accounted by diffusion.

8. Brownian Transport
It changes the relative position of molecules, but it does not carry volume. A displacement in one sense if balanced by another in the opposite sense. If is irrelevant for discharge.
It is relevant if the fluid is a mixture of molecules and concentration gradients are present . In that case the molecules moving in one sense are different from those moving in the opposite sense. Thus, there is a net mass transport.
Thus, the Brownian movement generates transport if there is a concentration gradient. The flux associated to this movement is a diffusive flux.
Fick realised that the diffusive flux is given by:
Where 𝜗 is the diffusivity
What is the diffusivity?

9. Diffusivity and Diffusion
Figures below represent 2 material systems, one fully white and the other fully Black separated by a diaphragm. The top figures represent the molecules (microscopic view) and the figures below the macroscopic view. When the diaphragm is removed the molecules from both systems start to mix and we start to see a grey zone between the two systems (b) at the end everything will be grey (c). During situation (b) we there is a diffusive flux of black molecules crossing the diaphragm section. This flux cannot be advective because velocity is null.
(a)
(b)
(c)

10. Ver texto sobre propriedades dos fluidos e do campo de velocidades
Diffusivity
When the diaphragm is removed molecules move randomly. The net flux is the diffusive flux.
The flux of molecules in each sense is proportional to the concentration and to the individual random velocity:
But,
Diffusivity is the product of the displacement length and the molecule velocity. This velocity is in fact the difference between the molecule velocity and the average velocity of the molecules accounted for in the advective term.
Ver texto sobre propriedades dos fluidos e do campo de velocidades

11. Diffusivity Diffusivity is definided as:
Where is the molecule velocity part not resolved (or included) in our velocity definition. In a laminar flow is the brownian velocity while in a turbulent flow is the turbulent velocity, a macroscopic velocity that we can see in the tubulent eddies.
is the lenght of the displacement of a molecule before being disturbed by another molecule (or of a portion of fluid in a turbulent flow). When the molecule hits another molecule it gets a new velocity.
Diffusivity dimensions are:

12. Molecular diffusivity, turbulent diffusivity, subgrid diffusivity
Molecular diffusivity is used if dA is of the order of the square of the free displacement of a molecule. It is a property of the fluid and can be displayed with other properties of the fluid.
Turbulent Diffusivity is used if the integration area is of the order of the largest eddy in the flow. It is a property of the flow (Reynolds Number) and no long a property of the fluid. In this case the length to be considered is the size of the eddy and the velocity is the random velocity.
Subgrid diffusivity is used in mathematical models. This models cannot resolve eddies smaller than the grid size and consequently the effect of those eddies has to be accounted through diffusion. Velocity is the average velocity across the faces of the control volume and diffusivity is computed as the product of the grid spatial step by the random velocity. 3D models using a fine resolution can resolve some eddies in the flow. So, the finer is the grid, the less important is the diffusion term.

13. The Advection – Diffusion Equation
{The rate of accumulation of a property inside a control volume} =
{what flows in minus what flows out}+{Production minus Consumption}
Figure 3‑4: Representation of a generic permeable surface, an elementary area on that surface, the respective exterior normal and the fluid velocity. dA

14. The rate of accumulationdA
Amount of property inside the volume:
The rate of accumulation is the rate of change of that quantity

15. The advective flux dA
Is the quantity of property flowing across a surface per unit of time:
Where B is the quantity (e.g. kg) and Vol is the volume (m3/s) and t is the time.
Defining β as the specific property (property per unit of volume) and the velocity as the volume per unit of time, one gets, per unit of area:
For an elementary area, one gets:
And for the whole area:

16. The diffusive flux The diffusive flux is given by:dA
The diffusive flux is given by:
For an elementary area, one gets:
And for the whole area:

17. The Advection – Diffusion Equation
{The rate of accumulation of a property inside a control volume} =
{what flows in minus what flows out}+{Production minus Consumption} dA
This equation was easy to obtain, it is easy to understand, but it is not useful to get results because the integrals very seldom have a solution.

18. Algebraic form of the Advection-Diffusion Equation
The integrals are easily calculated if the functions being integrated are constants along surfaces and the volume.
This is an acceptable assumption if surfaces are elementary, i.e. if their dimensions are much smaller than the scale of variation of the properties.
Assuming an elementary volume (with elementary faces) and a volume with faces perpendicular to the coordinated axis we get an algebraic equation.

19. Algebraic Equation
Constant in time

20. Summing up :
This equation in valid for a non-deformable elementary volume. An elementary volume is small enough for properties to be assumed uniform inside.

21. Differential Equation
The differential equation can be obtained shrinking the volume to zero (the side of the cube) and the time step top zero:
Summing up:

22. Differential Equation
Or,
If incompressible flow the divergence of the velocity is zero and the equation becomes:
If the control volume was moving at the flow velocity, the relative velocity would be zero and there would be no advection!!

23. Lagrangian approach
If the control volume was moving at the flow velocity, the relative velocity would be zero and there would be no advection!!
Using the definition of total derivative:
One would get:
Where the advective term does not appear!!!
This means that in the Lagrangian approach the time derivative is the rate of change of properties in a material portion of fluid, i.e. properties are being followed in a portion of fluid and not inside a region of space as in the Eulerian approach.

24. Summary
In this lesson we have seen different forms of the advection – diffusion equation and we have understood why there is always diffusion inside fluids.
The molecular diffusivity is a property of the fluid,
The turbulent diffusivity is a property of the flow (of the geometry and Reynolds number)
The sub-grid diffusivity is mostly a property of the grid used in the model.
Recommended reading: Notes about the advection-diffusion equation.