2. Advective Diffusion EquationIn nature, transport occurs in fluids through the combination of advection and diffusion. Theprevious chapter introduced diffusion and derived solutions to predict diffusive transport instagnant ambient conditions. This chapter incorporates advection into our diffusion equation(deriving the advective diffusion equation) and presents various methods to solve the resultingpartial differential equation for different geometries and contaminant conditions.2.1 Derivation of the advective diffusion equationBefore we derive the advective diffusion equation, we look at a heuristic description of the effectof advection. To conceptualize advection, consider our pipe problem from the previous chapter.Without pipe flow, the injected tracer spreads equally in both directions, describing a Gaussiandistribution over time. If we open a valve and allow water to flow in the pipe, we expect thecenter of mass of the tracer cloud to move with the mean flow velocity in the pipe. If we moveour frame of reference with that mean velocity and assume the inviscid case, then we expect thesolution to look the same as before. This new reference frame isη = x − (x0+ ut) (2.1)where η is the moving reference frame spatial coordinate, x0is the injection point of the tracer,u is the mean flow velocity, and ut is the distance traveled by the center of mass of the cloudin time t. If we substitute η for x in our solution for a point source in stagnant conditions weobtainC(x, t) =MA√4πDtexp −(x −(x0+ ut))24Dt!. (2.2)To test whether this solution is correct, we need to derive a general equation for advectivediffusion and compare its solution to this one.2.1.1 The governing equationThe derivation of the advective diffusion equation relies on the principle of superposition: ad-vection and diffusion can be added together if they are linearly independent. How do we knowif advection and diffusion are independent processes? The only way that they can be dependentis if one process feeds back on the other. From the previous chapter, diffusion was shown to bea random process due to molecular motion. Due to diffusion, each molecule in time δt will moveCopyrightc 2004 by Scott A. Socolofsky and Gerhard H. Jirka. All rights reserved.30 2. Advective Diffusion EquationJx,inJx,outx-yzδxδyδzuFig. 2.1. Schematic of a control volume with crossflow.either one step to the left or one step to the right (i.e. ±δx). Due to advection, each moleculewill also move uδt in the cross-flow direction. These processes are clearly additive and indepen-dent; the presence of the crossflow does not bias the probability that the molecule will take adiffusive step to the right or the left, it just adds something to that step. The net movement ofthe molecule is uδt ± δx, and thus, the total flux in the x-direction Jx, including the advectivetransport and a Fickian diffusion term, must beJx= uC + qx= uC − D∂C∂x. (2.3)We leave it as an exercise for the reader to prove that uC is the correct form of the advectiveterm (hint: consider the dimensions of qxand uC).As we did in the previous chapter, we now use this flux law and the conservation of massto derive the advective diffusion equation. Consider our control volume from before, but nowincluding a crossflow velocity, u = (u, v, w), as shown in Figure 2.1. Here, we follow the derivationin Fischer et al. (1979). From the conservation of mass, the net flux through the control volumeis∂M∂t=X˙min−X˙mout, (2.4)and for the x-direction, we haveδ ˙m|x=uC − D∂C∂x1δyδz −uC − D∂C∂x2δyδz. (2.5)As before, we use linear Taylor series expansion to combine the two flux terms, givinguC|1− uC|2= uC|1−uC|1+∂(uC)∂x1δx= −∂(uC)∂xδx (2.6)2.1 Derivation of the advective diffusion equation 31and− D∂C∂x1+ D∂C∂x2= −D∂C∂x1+D∂C∂x1+∂∂xD∂C∂x1δx= D∂2C∂x2δx. (2.7)Thus, for the x-directionδ ˙m|x= −∂(uC)∂xδxδyδz + D∂2C∂x2δxδyδz. (2.8)The y- and z-directions are similar, but with v and w for the velocity c omponents, givingδ ˙m|y= −∂(vC)∂yδyδxδz + D∂2C∂y2δyδxδz (2.9)δ ˙m|z= −∂(wC)∂zδzδxδy + D∂2C∂z2δzδxδy. (2.10)Substituting these results into (2.4) and recalling that M = Cδxδyδz, we obtain∂C∂t+ ∇· (uC) = D∇2C (2.11)or in Einsteinian notation∂C∂t+∂uiC∂xi= D∂2C∂x2i, (2.12)which is the desired advective diffusion (AD) equation. We will use this equation extensively inthe remainder of this text.Note that these equations implicitly assume that D is constant. When considering a variableD, the right-hand-side of (2.12) has the form∂∂xi Dij∂C∂xj!. (2.13)2.1.2 Point-source solutionTo check whether our initial suggestion (2.2) for a solution to (2.12) was correct, we substitutethe coordinate transformation for the moving reference frame into the one-dimensional versionof (2.12). In the one-dimensional case, u = (u, 0, 0), and there are no concentration gradients inthe y- or z-directions, leaving us with∂C∂t+∂(uC)∂x= D∂2C∂x2. (2.14)Our coordinate transformation for the moving system isη = x − (x0+ ut) (2.15)τ = t, (2.16)and this can be substituted into (2.14) using the chain rule as follows32 2. Advective Diffusion Equation0 1 2 3 4 5 6 7 8 9 1000.511.5Solution of the advective−diffusion equationPositionConcentrationt1t2t3CmaxFig. 2.2. Schematic solution of the advective diffusion equation in one dimension. The dotted line plots themaximum concentration as the cloud moves downstream.∂C∂τ∂τ∂t+∂C∂η∂η∂t+ u∂C∂η∂η∂x+∂C∂τ∂τ∂x=D∂∂η∂η∂x+∂∂τ∂τ∂x∂C∂η∂η∂x+∂C∂τ∂τ∂x(2.17)which reduces to∂C∂τ= D∂2C∂η2. (2.18)This is just the one-dimensional diffusion equation (1.29) in the coordinates η and τ with solutionfor an instantaneous point source ofC(η, τ ) =MA√4πDτexp −η24Dτ!. (2.19)Converting the solution back to x and t coordinates (by substituting (2.15) and (2.16)), weobtain (2.2); thus, our intuitive guess for the superposition solution was correct. Figure 2.2shows the schematic behavior of this solution for three different times, t1, t2, and t3.2.1.3 Incompressible fluidFor an incompressible fluid, (2.12) can be simplified by using the conservation of mass equationfor the ambient fluid. In