Questions about the numerical part of TEM (for test 2)

About the advection-diffusion equation

The advection-diffusion equation states that the rate of change of a property depends on the divergence of the advective and diffusive fluxes.

  1. The advective flux can be high and its contribution for the rate of change of a property can be negligible. Idem for the diffusive flux.
  2. Why does not radiation appear in the advection-diffusion equation?
  3. Why do we call “Heat equation” to the “Temperature evolution” equation? Why not write an equation for enthalpy instead of an equation for temperature?
  4. Why are diffusivity units m2/s for any property?
  5. Why is heat diffusivity higher than mass diffusivity in a fluid?
  6. Why are Metallic materials good heat conductors?
  7. Diffusivity is proportional to the product of the unresolved velocity (random) by the length of the free random walk. In case of molecular diffusivity, the random velocity is the velocity of a molecule and the length is the length of the free movement of that molecule. In turbulent flows the velocity is the eddy velocity and the length is proportional to the size of the largest eddies. In a numerical model the velocity is a fraction of the average velocity across a cell face and the length is the cell size.

About the numerical methods

  1. Numerical methods solve equations in an algebraic form. The physical principle is the same as that described by the corresponding partial differential equations. Why do algebraic equations have a certain ambiguity compared to differential equations?
  2. Errors in numerical models appear as “Numerical diffusion” or in terms of “numerical instabilities”. What is numerical diffusion and what is an instability?
  3. The courant number is an important parameter to understand numerical stability. What does it represent?
  4. In explicit methods the courant number cannot exceed 1. How is this related to stability?
  5. Why are implicit methods more stable than explicit methods?
  6. Are implicit methods more precise than explicit methods?
  7. Why do central differences for advection violate the transportive property of advection?
  8. Although central differences violate the transportive properties they can generate stable algorithms if the “grid Reynolds number” does not exceed 2. Why can diffusion stabilize central differences?
  9. Why can central differences generate negative concentrations?
  10. Is the upwind method always better than central differences?

About the boundary and initial conditions

  1. The boundary conditions are part of the solution (they are the solution along the boundary). Thus that mean that the solution is bad when we have uncertainty on boundaries?
  2. What kind of boundary conditions can we impose in numerical models? How do they compare with the boundary conditions imposed in the resolution of partial differential equations (Neumann-gradient/Dirichlet-imposed value).
  3. In a river it is common to assume that the heat flux across the bottom is null. This has some error. How does it influence the value of the computed water temperature? How could we improve this boundary condition?
  4. Evaporation is an important mechanism to reduce the temperature of a water body (or of any wet surface). What are the factors controlling it? Why is the wind velocity more important than the water velocity for the rate of evaporation?
  5. The correlations to compute sensible heat transfer and latent heat transfer are identical. In the former the temperature difference is the driving force, while in the latter the driving force is the gradient of water vapor partial pressure at the water temperature and at the atmospheric temperature and the transfer coefficient has the same shape. Why is this so?
  6. In case of gases surface fluxes also depend on the partial pressures, but the exchange coefficient is a function of water turbulence and is independent of the wind speed. Why is this so?
  7. Anthropogenic discharges can be of point or diffuse type. Point sources are usually easy to characterize, both in terms of flow rate and concentrations. Write the equation that gives the modification of concentration in the discharge cell due to a point discharge.
  8. What would you need to include a diffusive source of pollution in a river model (e.g. agriculture)
  9. Diffuse pollution is presently the major source of pollution in Europe (point sources are very much controlled). The excess of irrigation in agriculture enhances its impact on river water quality. How does this happen?

About the Results

  1. The 1D advection diffusion model can only simulate linear systems. It is not adequate to simulate ordinary lakes, but it could be used to simulate a long and narrow lake (e.g. a channel without flow). In this case the velocity would be null and there was no advection.
  2. How would an instantaneous emission in a cell evolve? What would be the final solution if there was no decay? And with decay?
  3. How would the solution evolve in that lake if there was a continuous emission without decay? Is it possible to get a stationary solution with decay?
  4. If there was velocity and no diffusion what should be the evolution? In reality there is numerical diffusion and the concentration will decay. Will the decaying rate due to numerical diffusion be the same along the whole channel?
  5. In case of a channel with advection and two point discharges, what would be the shape of the solution if there was no decay? And with decay?
  6. Legislation imposes maximum (or minimum) values for variables impacted by man, including for temperature and for biochemical pollutants. A treatment prior to discharge is necessary when the untreated discharge is too high. Show with an equation that the maximum values depend on size of the river (cross section) and on the river discharge.
  7. The temperature model shows a daily cycle. How does solar radiation contribute for the amplitude of that cycle? And sensible and latent heat fluxes?

About identical problems

  1. What would be the modifications of the model necessary to simulate:
  2. A solar oven?
  3. The evolution of the temperature in the ground, without rain
  4. The evolution of the temperature of the ground with rain
  5. The evolution of the temperature in a rectangular fin
  6. The evolution of the temperature in a triangular fin
  7. The temperature across a wall
  8. The temperature along a pipe