Investigation of scalar measurement error in diffusion and mixing processes

Abstract

In variable density, multi-fluid and reacting flows, the degree of molecular mixing is a critical component of turbulent transfer and mixing models. Also, in many microflows and low Reynolds number flows, scalar diffusion length- and time-scales play a significant role in the mixing dynamics. Characterization of such molecular mixing processes requires scalar measurement devices with a small probe volume size. Spatial averaging, which occurs due to finite probe volume size, can lead to errors in resolving the density or scalar gradients between pockets of unmixed fluids. Given a probe volume size and a priori knowledge of the functional profile of the diffusion layer being measured, we obtain an estimate for the measurement error due to spatial averaging and make the corrections accordingly. An analytical model for the measure of scalar mixing is developed as a predictor for the growth of scalar gradients in a variable scalar flow. The model is applied to a buoyancy-driven mixing layer with a Prandtl number of 7. Measurements within the mixing layer have shown that initial entrainment of unmixed fluid causes a decrease in the measured amount of molecular mixing at the centerplane. Following this period of initial entrainment, the fluids within the mixing layer exhibit an increase in the degree of molecular mixing.

Appendix A

1.1 Evaluation of convolution integral

The evaluation and algebraic simplification of the two convolution integrals in (14) are presented here for completeness. Given the actual scalar

$$\phi _{a} {\left(x \right)} = \ifmmode\expandafter\bar\else\expandafter\=\fi{\phi} + \frac{{\Delta \phi}}{2}\hbox{erf}{\left({\frac{x}{{L_{a}}}} \right),}$$

(30)

and the measurement probe’s response function

$${\Re}{\left(x \right)} = \left\{{\begin{array}{*{20}c} {{\frac{1}{{2R}}\hbox{e}^{{{x}/{R}}}}}, & {{x < 0}} \\ {{\frac{1}{{2R}}\hbox{e}^{{- {x}/{R}}}}}, & {{x \geqslant 0}} \\ \end{array}} \right. ,$$

(31)

the measured scalar trace is the convolution of the actual scalar value and response function

$$\phi _{m} {\left(x \right)} = \phi _{a} {\left(x \right)} \otimes R{\left(x \right)} = {\int\limits_{- \infty}^\infty {\phi _{a} {\left({x - {x}\ifmmode{'}\else$'$\fi} \right)}\,{\rm R}{\left({{x}\ifmmode{'}\else$'$\fi} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = {\int\limits_{- \infty}^\infty {\phi _{a} {\Re}{\left(x \right)}\,{\left({x - {x}\ifmmode{'}\else$'$\fi} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}}.$$

(32)

Both integrals in (3) produce the same result by the commutative property of the convolution (Balmer 1991), thus only the first integral will be evaluated.

Using the definitions of in (30) and (31), (32) can be rewritten as

$$\phi _{m} {\left(x \right)} = {\int\limits_{- \infty}^0 {{\left[{\ifmmode\expandafter\bar\else\expandafter\=\fi{\phi} + \frac{{\Delta \phi}}{2}\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}\,{\left[{\frac{1}{{2R}}\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} + {\int\limits_0^\infty {{\left[{\ifmmode\expandafter\bar\else\expandafter\=\fi{\phi} + \frac{{\Delta \phi}}{2}\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}\,{\left[{\frac{1}{{2R}}\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}}.$$

(33)

The integrals can be further divided into

$$\begin{aligned} \phi _{m} {\left(x \right)} &= {\int\limits_{- \infty}^0 {\ifmmode\expandafter\bar\else\expandafter\=\fi{\phi}\,{\left[{\frac{1}{{2R}}\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} + {\int\limits_{- \infty}^0 {{\left[{\frac{{\Delta \phi}}{2}\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}\,{\left[{\frac{1}{{2R}}\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}}\\ &\quad + {\int\limits_0^\infty {\ifmmode\expandafter\bar\else\expandafter\=\fi{\phi}\,{\left[{\frac{1}{{2R}}\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} + {\int\limits_0^\infty {{\left[{\frac{{\Delta \phi}}{2}\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}\,{\left[{\frac{1}{{2R}}\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}}. \end{aligned}$$

(34)

The integral of a constant $\bar{\phi}$ with ℜ(x) returns the value of the constant $\bar{\phi}$ because the integral of ℜ(x, R) over −∞ to ∞ is unity. Thus, the ϕ _m can be rewritten in a simpler form:

$$\phi _{m} {\left(x \right)} = \ifmmode\expandafter\bar\else\expandafter\=\fi{\phi} + \frac{{\Delta \phi}}{{4R}}{\int\limits_{- \infty}^0 {{\left[{\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}\,{\left[{\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} + \frac{{\Delta \phi}}{{4R}}{\int\limits_0^\infty {{\left[{\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}\,{\left[{\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}} \right]}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}}.$$

(35)

Symbolic integration of the two integrands in (35) were computed using Mathematica (Mathematica 2005):

$${\int\limits_{- \infty}^0 {\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = {\left[{- \hbox{Re}^{{{\left({\frac{{L^{2}_{a} + 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{{x}\ifmmode{'}\else$'$\fi}}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)} + \hbox{Re}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\, \hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}^{{{x}\ifmmode{'}\else$'$\fi = 0}}_{{{x}\ifmmode{'}\else$'$\fi = - \infty}},$$

(36)

$${\int\limits_0^\infty {\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{R}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = {\left[{R\hbox{e}^{{{\left({\frac{{L^{2}_{a} - 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{{x}\ifmmode{'}\else$'$\fi}}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)} - R\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\, \hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}} \right]}^{{{x}\ifmmode{'}\else$'$\fi = \infty}}_{{{x}\ifmmode{'}\else$'$\fi = 0}}.$$

(37)

The integration bounds will now be applied and each equation will be simplified. Starting with the (36),

$$\begin{aligned} {\int\limits_{- \infty}^0 {\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} &= {\left[{- R\hbox{e}^{{{\left({\frac{{L^{2}_{a} + 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{0}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)} + R\hbox{e}^{{{0}/{R}}} \hbox{erf}{\left({\frac{{x - 0}}{{L_{a}}}} \right)}} \right]}\\ &\quad - {\left[{- R\hbox{e}^{{{\left({\frac{{L^{2}_{a} + 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{- \infty}}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)} + R\hbox{e}^{{{{- \infty}}/{R}}} \hbox{erf}{\left({\frac{{x + \infty}}{{L_{a}}}} \right)}} \right]} .\\ \end{aligned}$$

(38)

Noting the following limits

$$\begin{aligned} &{\mathop {\lim}\limits_{x \to \infty}}{\left[{\hbox{erf }{\left(x \right)}} \right]} = 1 \\ &{\mathop {\lim}\limits_{x \to \infty}}{\left[{\hbox{erf }{\left({- x} \right)}} \right]} = - 1 \\ &{\mathop {\lim}\limits_{x \to \infty}}{\left[{\hbox{e}^{{- x}}} \right]} = 0\\ \end{aligned},$$

(39)

the first integral simplifies to

$${\int\limits_{- \infty}^0 {\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = {\left[{- R\hbox{e}^{{{\left({\frac{{L^{2}_{a} + 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)} + R\,\hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)}} \right]}\; - {\left[{- R\hbox{e}^{{{\left({\frac{{L^{2}_{a} + 4xR}}{{4R^{2}}}} \right)}}}} \right]}.$$

(40)

Combining terms and factoring gives

$${\int\limits_{- \infty}^0 {\hbox{e}^{{{{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{{L_{a}}}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = R\hbox{e}^{{{\left({\frac{{L^{2}_{a} + 4xR}}{{4R^{2}}}} \right)}}} {\left[{1 - \hbox{erf}{\left({\frac{x}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)}} \right]} + R\,\hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)}.$$

(41)

Applying the integration bounds to the second integral, (37), gives

$$\begin{aligned} {\int\limits_0^\infty {\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{R}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} &= {\left[{R\hbox{e}^{{{\left({\frac{{L^{2}_{a} - 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{\infty}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)} - R\hbox{e}^{{- {\infty}/{R}}} \hbox{erf}{\left({\frac{{x - \infty}}{{L_{a}}}} \right)}} \right]} \\ &\quad - {\left[{R\hbox{e}^{{{\left({\frac{{L^{2}_{a} - 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{0}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)} - R\hbox{e}^{{- {0}/{R}}} \hbox{erf}{\left({\frac{{x - 0}}{{L_{a}}}} \right)}} \right]}. \\ \end{aligned}$$

(42)

Using similar limit arguments used in the first integral, the second integral simplifies to

$${\int\limits_0^\infty {\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{R}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = {\left[{- R\hbox{e}^{{{\left({\frac{{L^{2}_{a} - 4xR}}{{4R^{2}}}} \right)}}}} \right]} - {\left[{R\hbox{e}^{{{\left({\frac{{L^{2}_{a} - 4xR}}{{4R^{2}}}} \right)}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)} - R\,\hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)}} \right]}.$$

(43)

Combining terms and factoring gives

$${\int\limits_0^\infty {\hbox{e}^{{- {{{x}\ifmmode{'}\else$'$\fi}}/{R}}}\,\hbox{erf}{\left({\frac{{x - {x}\ifmmode{'}\else$'$\fi}}{R}} \right)}\,\hbox{d}{x}\ifmmode{'}\else$'$\fi}} = - R\hbox{e}^{{{\left({\frac{{L^{2}_{a} - 4xR}}{{4R^{2}}}} \right)}}} {\left[{1 + \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)}} \right]} + R\,\hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)}.$$

(44)

Substituting the results from the simplification of each integral, (41) and (44), into (35) and simplifying, the measured scalar trace becomes

$$\begin{aligned} \phi _{m} {\left(x \right)} &= \ifmmode\expandafter\bar\else\expandafter\=\fi{\phi} + \frac{{\Delta \phi}}{4}{\left({\hbox{e}^{{{x}/{R}}} \hbox{e}^{{{\left({{{L_{a}}}/{{2R}}} \right)}^{2}}} {\left[{1 - \hbox{erf}{\left({\frac{x}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)}} \right]} + \hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)}} \right)} \\ &\quad + \frac{{\Delta \phi}}{4}{\left({- \hbox{e}^{{- {x}/{R}}} \hbox{e}^{{{\left({{{L_{a}}}/{{2R}}} \right)}^{2}}} {\left[{1 + \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)}} \right]} + \hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)}} \right)}.\\ \end{aligned}$$

(45)

Distributing Δϕ/4 and combining like terms results yields

$$\begin{aligned} \phi _{m} {\left(x \right)} &= \ifmmode\expandafter\bar\else\expandafter\=\fi{\phi} + \frac{{\Delta \phi}}{2}\hbox{erf}{\left({\frac{x}{{L_{a}}}} \right)} + \frac{{\Delta \phi}}{4}\hbox{e}^{{{\left({{{L_{a}}}/{{2R}}} \right)}^{2}}} {\left[{\hbox{e}^{{{x}/{R}}} - \hbox{e}^{{- {x}/{R}}}} \right]} \\ &\quad - \frac{{\Delta \phi}}{4}\hbox{e}^{{{\left({{{L_{a}}}/{{2R}}} \right)}^{2}}} {\left[{\hbox{e}^{{{x}/{R}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} + \frac{{L_{a}}}{{2R}}} \right)} + \hbox{e}^{{- {x}/{R}}}\, \hbox{erf}{\left({\frac{x}{{L_{a}}} - \frac{{L_{a}}}{{2R}}} \right)}} \right]}. \\ \end{aligned}$$

(46)

Replacing the first two terms with ϕ _a (x) [see (30)] and letting η _a = L _a /2R gives the final results shown in (18) and (19):

$$\phi _{m} {\left(x \right)} = \phi _{a} {\left(x \right)} + \frac{{\Delta \phi}}{2}\hbox{e}^{{\eta ^{2}_{a}}} \sinh {\left({\frac{x}{R}} \right)} - \frac{{\Delta \phi}}{4}\hbox{e}^{{\eta ^{2}_{a}}} {\left[{\hbox{e}^{{{x}/{R}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} + \eta _{a}} \right)} + \hbox{e}^{{- {x}/{R}}} \hbox{erf}{\left({\frac{x}{{L_{a}}} - \eta _{a}} \right)}} \right]}.$$

(47)