1. Evolution Equation Examples and 1D case Lecture 15 Mecânica de Fluidos Ambiental 2015/2016

2. Reynolds Theorem Mecânica de Fluidos Ambiental 2015/2016 Based on this theorem we have deduced that the difference between the total (or material) derivative and the local time derivative is the convective derivative: Reynolds Theorem states that:

3. Why does properties change inside a Material System? Due to Source and Sink terms (e.g. chemical reactions, biological activity) Due to diffusion in presence of concentration gradients Mecânica de Fluidos Ambiental 2015/2016

4. 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) there is a diffusive flux of black molecules crossing the diaphragm section. This flux cannot be advective because velocity is null. Mecânica de Fluidos Ambiental 2015/2016 Microscopic view Macroscopic view

5. Diffusivity Mecânica de Fluidos Ambiental 2015/2016 Ver texto sobre propriedades dos fluidos e do campo de velocidades But, Diffusivity is the product of the displacement length (“comprimento do deslocamento”) 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. 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:

6. Diffusivity Diffusivity is define as:  l.u b Where u b is the molecule velocity part not resolved (or included) in our velocity definition. In a laminar flow is the Brownian velocity in a turbulent flow is the turbulent velocity, a macroscopic velocity that we can see in the turbulent eddies.  l is the length of the displacement (“comprimento do deslocamento”) 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: L 2 T -1 Mecânica de Fluidos Ambiental 2015/2016

7. Diffusivity The diffusive flux is due to diffusivity and property gradient: The sense of the diffusive flux is opposite to the sense of the gradient. Diffusive flux is null if there is no gradient. Mecânica de Fluidos Ambiental 2015/2016

8. Diffusivity Gas – function of molecule properties (i.e. Fluid), molecule knectic energy (temperature), and the distance traveled by a molecule until collide with another (pressure and temperature). This diffusivity can be tabulated for each gas, fuction of tempeartura and pressure. Liquids – molecules are associated and form groups that move. Liquids are almost incompressibles, the difussivity only depend on the temperature, reducing the molecules group dimension in movement and therefore the difusivity. Mecânica de Fluidos Ambiental 2015/2016

9. E no caso da quantidade de movimento? Mecânica de Fluidos Ambiental 2015/2016 Se uma porção de fluido (e.g. molécula) desce da zona de maior velocidade para a de menor, vai aumentar a velocidade nessa zona. Nesse caso uma porção igual de fluido subirá e irá reduzir a velocidade em cima. Na presença velocidade aleatória e de gradiente de velocidades, o fluido mais rápido arrasta o mais lento. De acordo com a Lei de Newton, a uma aceleração corresponde uma força, que neste caso é uma força de atrito. À difusividade de quantidade de movimento chama-se viscosidade, que pode também ser vista como a relação entre a tensão de corte (atrito) e a taxa de deformação de um elemento de fluido (gradiente de velocidade). Escoamento com gradiente de velocidade.

10. Fluxo difusivo de Quantidade de Movimento e Tensão de Corte Mecânica de Fluidos Ambiental 2015/2016 O movimento aleatório não representado pela velocidade origina um fluxo de quantidade de movimento que é sentido como uma força (força de corte). Esta força aumenta com o gradiente de velocidade e depende da quantidade de massa que é necessário acelerar e da taxa a que a massa se move. τ(y) τ(y+Δy) Nesta equação as unidades da viscosidade (dinâmica) são (força/área)/segundo = >N/m2/s, Poiseuille no SI)

11. Evolution (or transport) Equation Mecânica de Fluidos Ambiental 2015/2016 Based on this theorem we have deduced that the difference between the total (or material) derivative and the local time derivative is the convective derivative: The Reynolds theorem states that:

12. Evolution Equation Mecânica de Fluidos Ambiental 2015/2016 Or:

13. Mecânica de Fluidos Ambiental 2015/2016 Diffusion plus advection Where is diffusive flux maximum? Diffusion

14. Analysis of the evolution equation Mecânica de Fluidos Ambiental 2015/2016 For the control volume: What is the sign of ? What does that mean physically (how does the advective flux vary with x 1 )? What is the relative value of diffusive flux in the lower and upper faces of the control volume? How does diffusive flux along x 2 contribute to the concentration inside the control volume? If the flow is stationary and the material is conservative what is the relation between advection and diffusion? what is the divergence of the total flux (advection + diffusion)? x1 x2

15. Answers Mecânica de Fluidos Ambiental 2015/2016 What is the sign of ? how does the advective flux vary with x 1 ? The advective term is negative. The velocity is positive and In other words, the advective flux decreases with x 1, i.e., the quantity entering is bigger than the quantity leaving (per unit of area). What is the relative value of diffusive flux in the lower and upper faces of the control volume? The diffusive flux in the lower face is about zero. If located exactly over the symmetry line, the flux will be exactly zero, because the gradient is null On the upper face the diffusive flux is positive, i.e., the material is transported along the axis x 2 because the concentration is higher inside the control volume than above the control volume.

16. Answers Mecânica de Fluidos Ambiental 2015/2016 How does diffusive flux along x 2 contribute to the concentration inside the control volume? The divergence of the vertical diffusive flux contributes to decrease the concentration inside the volume. If the flow is stationary and the material is conservative what is the relation between advection and diffusion? what is the divergence of the total flux (advection + diffusion)? The divergence of the horizontal diffusive flux is negative because the gradient on the left side of the volume is higher than on the right side. Horizontal advection is however the main mechanism to increase the concentration inside the control volume. If the flow is stationary the horizontal advection plus the horizontal diffusion balance the vertical diffusion. The divergence of both fluxes would be zero If the velocity was increased, the concentration inside the control volume would tend to increase because the quantity entering would increase and thus the quantity leaving would have to increase too.

17. Case of concentration Mecânica de Fluidos Ambiental 2015/2016 Consider two parallel plates and a property with a parabolic type distribution (blue) with maximum value at the center. Assume a stationary flow and a conservative property. a)Draw a control volume and indicate the fluxes (sense and relative magnitude). b)Where would the diffusive flux be maximum? Could this property be a concentration? What kind of property could it be? c)What is the sign of the material time derivative? d)If the material is conservative (no sink or source), can the profile be completely developed? a)Diffusive flux increase with “r”. Advective flux depends on the longitudinal gradient, not yet known. b)Property gradient is maximum at the boundary. This means that there the diffusive flux is maximum and consequently the property can pass through the boundary. The property can be a concentration only if there is adsortion at the boundary. It could be a temperature or momentum as well. c)Property is being lost across the boundary and consequently there is a longitudinal negative gradient, unless if there is a source. The total derivative is negative (and so is the advective derivative) d)No. Without a source there would be a negative longitudinal gradient. Otherwise it would depend on the source.

18. 1D case Mecânica de Fluidos Ambiental 2015/2016 Vamos supor o caso de um reservatório com a forma de um canal rectangular de 10 m de largura e 200 de comprimento, com velocidade nula. A concentração é elevada na zona central e nula na generalidade do canal. O material é conservativo. Escreva a equação que rege a evolução da concentração. E se existisse decaimento e não houvesse fontes?

19. With first order decay Mecânica de Fluidos Ambiental 2015/2016 How will the concentration evolve in each case? t0t0 t1t1 t2t2 t∞t∞ In case of decay, concentration are lower and tend to zero.

20. If there is a lateral discharge? Concentration will grow because of the discharge, will get homogenized because of diffusion and will decay because of decay. When discharge balances total decay, the system will reach equilibrium. Mecânica de Fluidos Ambiental 2015/2016

21. Solution of the problem The first case (pure diffusion) has analytical solution, but not the others. How to solve the problem numerically? Mecânica de Fluidos Ambiental 2015/2016

22. Let’s split the channel into elementary volumes Mecânica de Fluidos Ambiental 2015/2016 Cell i-1 Cell i Cell i+1 And apply the equation to each of them: If the property can be considered uniform inside each cell and along each surface : Where A is the area of the cross section between the elementary volumes, assumed constant in the academic example.

23. Numerical solution Mecânica de Fluidos Ambiental 2015/2016 Dividing all the equation by the volume: In this equation we have two variables,, and an extra variable, c, which time we have not defined. Are the actual concentrations and are the concentrations that we want to calculate. The concentration c is the concentration used to calculate the fluxes and decay. What is it?

24. Temporal discretization The flux equation gives quantity per unit of time. When equation as: We are computing the amount that crossed the surface during a time period (  t) and dividing it by the length of the time. The most convenient time to allocate to c is the middle of the time interval: Mecânica de Fluidos Ambiental 2015/2016

25. The equation becomes: Mecânica de Fluidos Ambiental 2015/2016 This equation has 3 unknowns. The solution for all channel requires the resolution of a system of equations in each time step.

26. Explicit calculation Mecânica de Fluidos Ambiental 2015/2016 That can be solved explicitly, but has stability problems. If we had assumed : We would have got:

27. Implicit calculation Mecânica de Fluidos Ambiental 2015/2016 That has less computation than the “average” approach, but still requires the resolution of a system of equations. If we had assumed: We would have got:

28. Results (D<0.5) Mecânica de Fluidos Ambiental 2015/2016 D=0.25 DT=100 dx20 difusividate1 k=0 Timei-3i-2i-1ii+1i+2i+3 0000010000 1000000.250.50.25000 200000.060.250.380.250.0600 30000.020.090.230.310.230.090.020 40000.030.110.220.270.220.110.030 5000.010.030.120.210.250.210.120.040.01 The results evolve as expected.

29. Results (D=0.75) Mecânica de Fluidos Ambiental 2015/2016 D=0.75 DT=300 dx20 difusividate1 k=0 Timei-4i-3i-2i-1ii+1i+2i+3i+4 0000010000 3000000.75-0.50.75000 600000.56-0.81.38-0.80.5600 90000.42-0.81.83-1.81.83-0.80.420 12000.32-0.82.11-2.93.65-2.92.11-0.80.32 1500-0.790.71-3.95.77-6.25.77-3.90.71-0.79 The results are strange. Concentration bigger than initial are obtained. Negative concentrations are also obtained…. The method is unstable

30. Stability The method became unstable because the parenthesis becomes negative when D>0.5; When the parenthesis changes sign, the effect of c i at time t changes: If it is positive, the larger is the initial concentration the larger is the subsequent concentration. If it is negative, the larger is the initial concentration the smaller is the subsequent, which is physically impossible. This is the limitation of the explicit methods. They are conditionally stable. In pure diffusion problems the D (the diffusion number) must be smaller than 0.5. If other processes exist (e.g. k≠0) D must be smaller. Mecânica de Fluidos Ambiental 2015/2016

31. Implicit methods stability Implicit (as well as semi-implicit) methods are more difficult to program, but do not have stability limitations allowing larger values for D, meaning that we can combine small grid sizes with large time steps, while in explicit methods the time step is associated to the square of the spatial step. Mecânica de Fluidos Ambiental 2015/2016

32. How large should diffusivity be? We have to reassess the concept of diffusivity: “Diffusivity is the product of a non-resolved velocity part by the length of the displacement of the fluid due to that velocity”. Mecânica de Fluidos Ambiental 2015/2016

33. Value of diffusivity in our problem When we assume the velocity to be uniform in the cross section area, we are neglecting spatial variability. If the profile was parabolic the average velocity would be half of the maximum velocity. We would be neglecting the eddies with velocity similar to the average velocity and radius of half channel width. Diffusivity should be of the order of: UL/2. Mecânica de Fluidos Ambiental 2015/2016

34. Stability The system gets unstable if D>0.5 This equation shows that the length of the displacement due to diffusion must be smaller than half of the cell length. Mecânica de Fluidos Ambiental 2015/2016

35. Initial Conditions Initial values to be provided in each cell Mecânica de Fluidos Ambiental 2015/2016

36. Boundary conditions Boundary conditions specify how the fluid interacts with the surrounding environment. Solid boundaries Open boundaries, In geophysics boundaries can be vertical of horizontal (Top and bottom). Gravity make them important. One can impose property’s values or fluxes (advection and/or diffusion). Mecânica de Fluidos Ambiental 2015/2016

37. Input data Geometry, Flow properties, Process parameters, Execution parameters. Mecânica de Fluidos Ambiental 2015/2016