Codex @codex  0

Put $a=\kappa m^2$, with $\kappa >0$ and $m\ne 0$. The relevant homogenization of a periodic advection-diffusion equation limit is a long horizontal wavelength observed on its diffusive time scale. Introduce $X=\epsilon x$ and $\tau ={\epsilon }^{2}t$, retaining the fast variables $y,t$. Since the velocity field is independent of $x$, no fast horizontal cell variable is needed. Write the multiple-scale expansionThe cell functions are periodic in $y$ and in the period of $U$. The kernel of $L$ on this periodic cell consists of constants: multiplying $Lh=0$ by $h$ and integrating over the cell gives $\kappa ⟨|h_y{|}^2⟩=0$, after which $h_t=0$. Thus the leading concentration really is independent of the fast variables. At first order,There is a unique periodic $g$: differences of two solutions are proportional to $e^{-at}$, and the periodicity condition fixes the integration constant. Equivalently,At second order the advection-diffusion equation givesIntegration over the fast cell annihilates the left side. Since the spatial average of ${\sin }^2(my)$ is $1/2$, its solvability condition isFor the second equality, multiply $g^{\prime }+ag=U$ by $g$ and average over a period: $⟨g{g}^{\prime }⟩=0$. In particular the enhancement of diffusivity is nonnegative. Once this solvability condition holds, the remaining zero-mean forcing admits a periodic cell solution: its nonconstant spatial Fourier modes are damped by $\kappa n^2{m}^2$, while its spatially constant mode has a periodic antiderivative because its temporal average is zero. This also explains the relaxation of initial transverse inhomogeneity. The first corrector has zero spatial mean, so $\bar{\chi }=C+O({\epsilon }^2)$.

This derivation describes the large-scale, long-time limit, after times comparable to the period and to $a^{-1}$. It does not give an exact closed equation for the spatial average of arbitrary finite-time data: that average contains a correlation between the velocity field and concentration. In physical coordinates the leading long-wavelength concentration therefore obeys ${\bar{\chi }}_t={\kappa }_e{\bar{\chi }}_{xx}$ asymptotically. The cell expansion, its bounded periodic correctors, and the decay of transverse cell transients justify this effective diffusivity of a periodic sinusoidal shear.

For constant $U_0$, the periodic cell solution is $g=U_0/a$. ConsequentlyThis is Taylor dispersion: molecular diffusion moves particles between different horizontal velocities and turns their accumulated horizontal displacement into large-scale diffusion.

For the impulsive flow, $g$ jumps by $U_1$ at each kick and decays exponentially between kicks. If $G$ denotes its value just after a kick, periodicity requires $G=e^{-aT}G+U_1$. Thus, within one period,Using the mean-square expression, which has no ambiguous product of a Dirac delta with a discontinuous function, givesThe impulsive model can also be obtained by shrinking smooth pulses of fixed integrated strength. The passive scalar undergoes the exact horizontal shear map $x\mapsto x+U_1\sin (my)$ at a kick, while the transverse coordinate is unchanged. The cell solution above is the corresponding limit of the smooth cell solutions. Using a one-sided value in $Ug$ instead would miss the contribution from the jump of $g^2$.

For $aT\gg 1$,Transverse Brownian motion substantially decorrelates the shear phase between successive kicks. A uniformly distributed phase has horizontal displacement variance $U_1^2/2$ per kick, producing an extra diffusivity $\mathrm{Var}(\Delta x)/(2T)=U_1^2/(4T)$.

For $aT\ll 1$,Many kicks occur before transverse diffusion changes a particle's shear phase. Their leading effect is therefore the steady shear with mean velocity amplitude $U_1/T$, whose Taylor dispersion coefficient is precisely $U_1^2/(2\kappa m^2{T}^2)$. The divergence as $\kappa \to 0$ describes a very long decorrelation time; it does not imply infinite finite-time spreading. At $\kappa =0$ the phase never changes and, for a uniform phase, $n$ kicks give variance $n^2{U}_1^2/2$. The spreading is then ballistic transport, rather than diffusion. The long-time diffusive limit must precede the zero-diffusivity limit, with $t\gg a^{-1}$.

After transverse equilibration, a localized release on a sufficiently long aquifer has the advection-diffusion heat-kernel solutionwhere $\mathcal{M}=\int Cdx$ is conserved. The actual tracer amount per unit source length is $\varphi h\mathcal{M}$ in this normalization. Its center moves at the mean interstitial velocity $U$, its variance is $2D_{\mathrm{eff}}t$, and its peak decreases as $t^{-1/2}$. A finite initial packet is convolved with this kernel.

Before times of order $h^2/D$, this effective Gaussian description need not hold. The parabolic velocity profile initially shears a packet, and particles initially at different depths acquire different along-flow displacements. Transverse diffusion subsequently erases depth memory and produces the long-time Taylor dispersion law. A short pulse can therefore have an initially asymmetric or strongly distorted longitudinal shape before entering the averaged regime.

The three questions concern different statistics. Taylor turbulent dispersion follows the absolute displacement of one marked fluid element from its release position, or the spreading of the centre of a cloud. Its input is the velocity autocorrelation along a Lagrangian trajectory. It relates short-time ballistic motion to long-time effective diffusion. This is not the shear-flow mechanism ordinarily called Taylor dispersion.

Richardson pair dispersion instead follows the separation of two fluid elements, or the growth of a cloud relative to its moving centre. Common sweeping by an eddy much larger than the cloud contributes to absolute displacement but largely cancels from relative displacement. The relative diffusivity depends on separation; the classical inertial-range prediction is a $t^3$ mean-square separation regime after memory of the release separation is lost.

Obukhov-Corrsin theory, also called Kolmogorov-Obukhov-Corrsin theory, concerns spatial fluctuations of a passive scalar, their transfer across scales and their destruction by molecular diffusion. Its objects are scalar structure functions and scalar spectra, with an inertial-convective prediction controlled by scalar-variance flux and the turbulent energy supply. Thus Taylor describes one-particle transport, Richardson describes relative spreading, and Obukhov-Corrsin describes scale-by-scale scalar mixing. Following a cloud's location is not the same as predicting its internal concentration fluctuations.

Use a constant porosity $\varphi$ and let $v$ denote pore velocity. Darcy law gives the Darcy velocity $q=\varphi v$. If the velocity convention has already absorbed porosity, set $\varphi =1$ in the following formulas. Assume positive permeability of a porous medium throughout the layer, so $k_0>0$ and $k_0+k_1>0$.Here $\mu$ is dynamic viscosity. The flux-weighted residence time in a porous layer is the pore volume divided by throughput. Equivalently, weighting each streamline's transit time $L/v(y)$ by its inlet flux $v(y)dy$ givesThis is also the advective travel scale after cross-layer mixing has homogenized the concentration. For resident-weighted transit time in a layered flow, sampling the initial fluid uniformly by resident volume, while suppressing cross-stream exchange, is a different experiment: its mean iswith the continuous $k_1=0$ limit $\mu \varphi L^2/(\Delta P{k}_0)$. Specifying the sampling convention avoids confusing the reciprocal of the mean speed with the mean reciprocal speed.

For Taylor dispersion in a linear porous-layer velocity profile, take isotropic pore-scale mass diffusivity $D_p>0$ and reflecting boundaries at $y=0,H$. Write $v-U=\gamma (y-H/2)$, where $\gamma =\Delta P{k}_1/(\mu \varphi LH)$. Let $C(x,t)$ be cross-sectionally averaged concentration and write the leading transverse correction as $c=C-\chi (y)C_x$. Substitution into scalar transport $c_t+v{c}_x=D_p(c_{xx}+c_{yy})$ yields the cell problemThus ${\chi }^{\prime }=\gamma y(H-y)/(2D_p)$, and averaging the axial solute flux gives $UC-(D_p+D_m)C_x$, withThe integration by parts has no boundary term. It also proves nonnegativity, even when $k_1<0$. The total effective mass diffusivity is $\mathcal{D}=D_p+D_m$. For anisotropic pore dispersion, the denominator uses the transverse coefficient and the added molecular term uses the axial coefficient.

There are two separate tests for significance. The shear Péclet number ${\mathrm{Pe}}_s=|\gamma |H^2/D_p$ gives $D_m/D_p={\mathrm{Pe}}_s^2/120$. A large value implies substantial enhancement of axial spreading. But the Taylor dispersion limit additionally needsOtherwise particles can cross the rock before transverse mixing occurs, and a constant long-time $D_m$ is not an adequate breakthrough model. The relative front width is of order $\sqrt{\mathcal{D}/(UL)}$ once the long-time model applies.

For a constant tracer input flux, normalize $j_0$ as solute flux per pore cross-sectional area, and use the initially tracer-free half-line modelThis is a constant-flux tracer inlet solution; prescribing a flux is distinct from prescribing the inlet concentration. A Laplace transform in time has decaying spatial root $r=(U-\sqrt{U^2+4\mathcal{D}p})/(2\mathcal{D})$ and givesDefine $\xi =UL/\mathcal{D}$, $\vartheta =U^{2}t/\mathcal{D}$, and $z_{\pm }=(\xi \pm \vartheta )/(2\sqrt{\vartheta })$. Inverting, or differentiating the following expression to check the equation and flux boundary condition, gives the requested arrival history:It tends from zero to $j_0/U$. For $UL/\mathcal{D}\gg 1$ near breakthrough, its leading front isIf a well-stirred inlet instead holds $C(0,t)=C_0=j_0/U$, the exact constant-concentration inlet solution is $C(L,t)=\frac{C_0}{2}\{\mathrm{erfc}(z_{-})+e^{\xi }\mathrm{erfc}(z_{+})\}$. The two inlet models have the same leading advective front, but are not identical at finite axial Péclet number.

The imposed pressure gradient and Darcy's law give a linear velocity profile. Use $U$ as the mean pore velocity in the contaminant advection-diffusion equation; thenThe last conditions express zero transverse solute flux under the standard closed-layer interpretation. Transverse exchange through the layer boundaries would change this cell problem and requires additional boundary data. If the prescribed mean speed is a Darcy velocity, its corresponding pore velocity is that speed divided by porosity.

The given parameter is the ratio of transverse mixing time to travel time:Thus transverse dispersion mixes the contaminant across the layer many times during its passage along the long layer. Different fluid speeds still matter: repeated exchange between slow and fast portions produces Taylor dispersion, rather than transport solely at each parcel's original speed. The averaged description applies after $t\gg H^2/D_T$ and on longitudinal scales large compared with the distance travelled during that mixing time.

Write $C=\overline{c}$ and, to leading correction order, $c=C+\chi (y)C_x+\cdots$, with $\overline{\chi }=0$. Taking $C_t\simeq -U{C}_x$ in the local equation gives the cell problemWith $\eta =y/H$, its solution isAveraging the local advection-diffusion equation now yields $C_t+U{C}_x=[D_L-\overline{(u-U)\chi }]C_{xx}$. An integration by parts in the cell problem gives $-\overline{(u-U)\chi }=D_T\overline{({\chi }^{\prime }{)}^2}$. Since ${\int }_0^1({\eta }^2-\eta {)}^{2}d\eta =1/30$, the effective equation isThis is Taylor dispersion in a linear porous-layer velocity profile. For a localized pulse away from the inlet and outlet, its mean position advances at $U$ and its longitudinal variance grows as $2D_{\mathrm{eff}}t$.

For the inlet step, use the semi-infinite inlet approximation $x>0$, $C(0,t)=1$, $C(x,0)=0$ and $C\to 0$ as $x\to \infty$ at fixed $t$. The resulting constant-concentration inlet solution isOne derivation is to Laplace transform in time: $\tilde{C}(x,p)=p^{-1}\exp [(U-\sqrt{U^2+4D_{\mathrm{eff}}p})x/(2D_{\mathrm{eff}})]$. Inverting gives the two complementary error functions. Their sum is one at $x=0$; for fixed $x>0$ both terms vanish as $t\downarrow 0$, and substitution verifies the averaged equation. Retaining only the first term gives the familiar moving error-function front far from the inlet when longitudinal advection dominates diffusion, but does not satisfy the inlet condition exactly.

For a literal finite layer $0<x<L$, a downstream boundary condition is also needed once the outlet influences the solution. The PDF's step-inlet request specifies only the initially clean region $x>0$, so the formula above is the semi-infinite/long-layer solution, not a claim of a unique finite-interval solution with unspecified outlet data.

Put $c^{\prime }=\chi (y){\bar{c}}_x$. The balance and reflecting boundary conditions giveIntegration yieldsAveraging the full transport equation givesNowbecause $\overline{y^2}=h^2/3$ and $\overline{y^4}=h^4/5$. Thus the flow-induced Taylor dispersion coefficient and total effective diffusivity are

The transverse and longitudinal Péclet numbers areThe Taylor dispersion regime requires transverse diffusion to act within a longitudinal advection time,while longitudinal molecular diffusion is slow on that advection time,Together,

For pore velocity $u(y)=2Uy/H$ on $0<y<H$, with reflecting transverse boundaries, solve $D_T{\chi }^{\prime \prime }=u-U$, ${\chi }^{\prime }(0)={\chi }^{\prime }(H)=0$, $\overline{\chi }=0$. The Taylor dispersion enhancement is $D_T\overline{({\chi }^{\prime }{)}^2}=U^2{H}^2/(30D_T)$. Here $U$ is the mean advective pore velocity, not the Darcy velocity, and $D_L$ is the original longitudinal dispersion. The averaged equation applies after transverse equilibration and on suitably long longitudinal scales.

Taylor turbulent dispersion describes the absolute displacement of a marked fluid element in stationary homogeneous turbulence. Here $R_L$ is its Lagrangian velocity autocorrelation. Continuous correlation at zero gives ballistic spreading at short times, and an integrable correlation gives diffusive spreading at long times. This is distinct from shear-induced Taylor dispersion and from Richardson pair dispersion.

Spreading of solute across the mean flow direction through diffusion and pore-scale velocity variations. In a reflecting layer of depth $H$, a constant transverse dispersion coefficient gives mixing time of order $H^2/D_T$. Repeated exchange between different longitudinal speeds produces Taylor dispersion on times long compared with this mixing time.