A modified two-layer scalar diffusivity description for high Schmidt and Prandtl turbulent boundary layers

Wong, Kin Wing; Mickus, Ignas; Grishchenko, Dmitry; Kudinov, Pavel

doi:10.1063/5.0255551

In engineering systems operating under high Schmidt (Sc) or Prandtl (Pr) number flow conditions, the demand for near-wall mesh refinement increases significantly, underscoring the need for cost-effective modeling approaches that avoid additional computational overhead. Existing models, which are predominantly designed for low-Sc flows, overlook temporal filtering effects, resulting in inaccuracies in theoretical description and mass transfer predictions. This paper addresses the impact of high Sc or Pr by refining the single-layer scalar diffusivity model. It introduces a switch between scalar filtering and eddy viscosity-dominated regions, leveraging two parameters: $κ$ _Sc, accounting for temporal filtering effects, and $κ$ _Re, addressing variations in Reynolds number. In addition, we adopted a complementary outer layer term to model the upwarding trend in low frictional Reynolds number condition. Using the two-layer model with unity Sc and/or Pr, a close agreement with the von-Kármán constant in the velocity boundary layer was observed. The modified model demonstrated strong agreement with scalar profiles across a broad range of Sc and friction Reynolds numbers (Re_τ) in direct numerical simulation and large eddy simulation data, demonstrating its accuracy at low Re_τ and predictive performance at high Re_τ. The two-layer model improves the prediction of turbulent mass transfer, providing better alignment with high Sc engineering correlations than existing wall model approach. This study provides valuable insight for modeling the mass and heat transfer processes under high Sc or Pr conditions.

NOMENCLATURE

$C_{μ}, f_{μ}$: Empirical functions in $k - ϵ$ model
$c$: Passive scalar
$c^{+}$: Dimensionless scalar
$c_{τ}$: Frictional scalar
$f$: Friction factor
$L_{cond}$: Conductive boundary layer thickness (m)
K_v: Mass transfer coefficient (m/s)
$k$: Turbulence kinetic energy (m²/s²)
Pr: Prandtl number
$p$: Pressure (Pa)
$R e_{τ}$: Frictional Reynolds number
$R e$: Reynolds number
Sc: Schmidt number
Sh: Sherwood number
St: Stanton number
$u_{i}^{'} c^{'}$: Scalar flux [kg/(m² s)]
$u_{i}^{'} u_{j}^{'}$: Reynolds stress (m²/s²)
$u_{j}$: Velocity (m/s)
$u_{τ}$: Frictional velocity
$y^{+}$: Dimensionless wall distance
$α$: Molecular diffusivity (m²/s)
$α^{+}$: Dimensionless scalar diffusivity
$α_{eff}$: Effective diffusivity (m²/s)
$α_{t}$: Turbulent diffusivity (m²/s)
$β$: Switching function in two-layer model
$γ$: Exponent in two-layer model
$κ_{Re}$: Re-dependent function in two-layer model
$κ_{Sc}$: Sc-dependent function in two-layer model
$μ$: Dynamic viscosity (Pa s)
$ν$: Kinematic viscosity (m²/s)
$ν_{t}$: Turbulent viscosity (m²/s)
$ρ$: Density (kg/m³)
$τ$: Wall shear stress (N/m²)
$ω$: Turbulence dissipation rate (1/s)
$ϵ$: Turbulence dissipation (m²/s³)

Acronyms

DNS: Direct numerical simulation
LES: Large eddy simulation
RANS: Reynolds-averaged Navier–Stokes
RSM: Reynolds stress model
SEFACE: Separated effect flow accelerated corrosion and erosion
SGDH: Simple gradient diffusion hypothesis
SST: Shear-stress transport
WM: Wall-modeled
WMLES: Wall-modeled LES
WR: Wall-resolved

I. INTRODUCTION

Modeling passive scalar transport in turbulent flows is critical for predicting heat and mass transfer processes in a wide range of industrial applications.^1–5 Among these, flow-accelerated corrosion (FAC) stands out as a key challenge due to its significant impact on system reliability and maintenance costs. To improve our understanding of FAC, accurate modeling of the turbulent mass transfer process is required. Particularity, the Schmidt (Sc) numbers relevant to FAC, ranging from a few hundred to several thousands in water and liquid metal medium (as discussed in Chen et al.⁶ and reader can consult the thermophysical properties handbook from OECD and Nuclear Energy Agency⁷ for further details in liquid metal community), exemplify the complex dynamics of scalar transport, characterized by steep concentration gradients in complex turbulent boundary layer.

Despite advances in computational fluid dynamics (CFD) and experimental techniques, significant gaps remain to simulate these processes under industrially relevant conditions. Current limitations include a lack of validation data across the parameter space encountered in real-world applications, as well as a scarcity of direct numerical simulation (DNS) datasets for high Sc flows at the condition relevant for industrial applications. These challenges are compounded by the difficulty of resolving the passive scalar boundary layer, where scalar and momentum transport are tightly coupled, without prohibitive computational costs.

A critical need is the development of reliable wall models to bridge the fidelity gap between DNS and practical engineering simulations. These models must accurately capture near-wall scalar transport phenomena, while maintaining the computational efficiency required for routine use in engineering design. Most of the mass transfer applications involved the development of phenomenological models, such as dissolution or/and oxidation models. Minimizing the uncertainty in fluid-mass transfer predictions can help to access the accuracy of phenomenological models in a non-biased manner.

Several studies on turbulent heat and mass transfer using large eddy simulation (LES) and DNS at high Sc or Pr have been reported in the open literature. Schwertfirm and Manhart⁸ performed DNS calculation of a channel flow using a two-grid approach under a Re_τ = 180 and a Sc range from 1 to 49. Reeuwijk and Hadžiabdić⁹ performed DNS of a channel flow under a Re_τ = 400 and a Sc range up to 50. Balasubramanian et al.¹⁰ performed DNS calculation for zero pressure gradient boundary layer with Re_τ from 420 to 1070 and Pr up to 6. Pirozzoli¹¹ performed DNS calculation of turbulent pipe with Re_τ at Pr up to 16. To the best of the authors' knowledge, there is no further Eulerian DNS study for Sc higher than 50 reported in open literature.

On the other hand, Lagrangian simulations can be employed to obtain DNS data at significantly higher Sc or Pr values up to the O(10⁴), as demonstrated in the work of Papavassiliou and Hanratty,¹² Na and Hanratty,¹³ and Mito and Hanratty.¹⁴ However, these simulations primarily focus on the lower range of Re_τ around 150 or Re of 2660. Notably, Hasegawa et al.¹⁵ performed hybrid LES-DNS calculation of a channel flow under a Re_τ = 150 and Sc up to 400. Similarly, Bergant and Tiseji¹⁶ conducted LES investigation channel flow under a Re_τ = 150, 350 and Sc up to 500. Due to the significant computational demands, Eulerian calculations at high Re_τ and high Sc remains scarce in the literature. In contrast, there is a wealth of DNS data available in the open literature for unity Sc or Pr numbers from low to high Re_τ, including studies such as Schwertfirm and Manhart,⁸ Kozuka et al.,¹⁷ Pirozzoli,¹⁸ and Pirozzoli et al.¹⁹ This paper seeks to propose an approach of leveraging the limited high Sc DNS data to enhance models tailored for lower Sc ranges.

Several approaches of wall model/function are documented in the open literature. Given that passive scalar transport cannot be characterized by a single scale, Jayatilleke²⁰ introduced an empirical P-function approach to model the logarithmic region of the thermal boundary layer. By integrating temperature profiles from turbulent flow in pipes and channels containing liquid metals, air, water, ethylene glycol, and technical oils, with a Sc range from 0.7 to 170 and a Re on the order of O(10⁴), Kader²¹ proposed an empirical law that has become a standard reference for wall modeling of passive scalars, however, lacking the fundamental consideration of the filtering effect in high Sc scenarios.

With the advent of modern computational architecture, wall models derived from DNS have been reported in the open literature. One notable example is the work of Schwertfirm and Manhart,⁸ who derived the logarithmic law based on DNS data for channel flow up to a Sc of 49. Another significant contribution is from Pirozzoli,¹¹ who suggested that, according to their DNS data for Pr up to 16 at high Re_τ, the thermal (scalar) diffusivity exhibits a trend similar to that of the turbulent kinematic viscosity (⁠ $ν_{t}$ ⁠) reported by Musker²² (i.e., the near-wall turbulent viscosity and scalar diffusivity follows the y⁺³ scaling at the near wall region), with the lower-order terms in the numerator being truncated. Pirozzoli¹¹ is considered to be the state-of-the-art in scalar diffusivity wall model for variable Pr number flows. However, none of the wall models mentioned above fully encompass the entire range of Sc relevant to FAC applications.

It is crucial to extend the models to higher Sc ranges to cover the interplay between Sc effects and the flow dynamics. For example, the Sc of elemental constituents can vary significantly in an order of magnitude from the hot-leg to cold-leg conditions and reader can consult the thermophysical properties database by the OECD and Nuclear Energy Agency⁷ and the measurement and the molecular dynamics simulation of Fe and Ni diffusion in liquid lead-bismuth by Gao et al.^23,24 for more details. In this study, we aim to enhance the state-of-the-art scalar-diffusivity based wall model—specifically, the Pirozzoli¹¹ approach, such that the near-wall diffusivity behavior is modeled correctly to allow its extension to high Sc conditions.

Apart from the explicit wall models mentioned above, Suga et al.²⁵ modeled high Pr turbulent flows using an analytical wall model approach. This method modeled the thermal diffusivity within the sub-layer (for y⁺ < 11.7) by employing the y⁺³ power law, similar to Pirozzoli.¹¹ However, the approach in Suga et al.²⁵ requires solving the parallel momentum and scalar equations in the near-wall region, which limits its applicability in the open literature. In this paper, we focus on developing a simple modification that can be readily applied in existing CFD code without extra implementation and computational burden.

In modeling high-Sc turbulent flows, a key limitation in above-mentioned approaches is the ignorance of the temporal filtering effect within the conductive boundary layer of the passive scalar field. As a result, deviations from the commonly assumed y⁺³ power law in the conductive boundary layer are observed in experimental and DNS studies. This occurs because the assumption of Reynolds analogy breaks down in high Sc flows as hinted by Mitrovic et al.²⁶ These discrepancies become particularly pronounced at large Sc, where the impact of incorrect near-wall scaling becomes increasingly significant. An example can be seen in LES studies of high-Sc turbulent flows from early work by Calmet and Magnaudet²⁷ and Dong et al.,²⁸ where the scaling relationship is not preserved due to insufficient near-wall grid resolution or the inaccurate subgrid scalar flux modeling. As a result, the filtering effects are only observable in DNS calculations. This limitation is not confined to LES; existing Reynolds-averaged Navier–Stokes (RANS) models also face challenges in accurately resolving the near-wall scaling due to incorrect scalar flux modeling for high-Sc flows.

Wong et al.²⁹ demonstrated in that the thermal perturbations induced by sliding bubbles in a high Pr laminar flow require a longer time to revert to the laminar state than the velocity perturbations. For turbulent flow, similar effects occur and lead to distinct scaling in the turbulent energy spectrum for velocity and passive scalar fields as discussed by Hasegawa et al.³⁰ The lower diffusivity of the passive scalar field compared to the velocity field under high Sc conditions results in distinct response timescales for the passive scalar and velocity fields.

Figure 1(a) illustrates the different responses of the passive scalar field with varying diffusivities, for “low Sc diffusivity” (same order as momentum diffusivity) and “high Sc diffusivity” (much smaller than viscosity) diffusivity to a turbulence-like signal. The low Sc diffusivity response follows closely to the turbulent-like input signal, therefore, for such a case, there is only a restricted impact of temporal filtering. Since the effect of low Sc diffusivity only lies in the reduction of the peak amplitude of the turbulence like signal, in other words, the scalar profile tracks closely to the velocity signal (i.e., Reynolds analogy might hold). On the other hand, the high Sc diffusivity response exhibits not only a reduction of the response amplitude but also a time-lag due to the slow response time of the scalar signal (i.e., Reynolds analogy might not hold). Figure 1(b) compares the statistical distributions for “low Sc” and “high Sc diffusivity” diffusivity cases against the turbulence-like velocity signal in the form of a histogram. The extent of filtering of the spectrum is stronger in Fig. 1(b) for the high Sc than the low Sc diffusivity case.

FIG. 1.

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Schematic comparison of response time of velocity and passive scalar fields.

A similar analogy can be drawn for near-wall scalar fluctuations, where the time-lag effect in the decay of passive scalar fluctuations can significantly influence the behavior of scalar eddy diffusivity near the wall. The temporal filtering effect, on scalar fluctuations in turbulent flows was investigated by Hasegawa and Kasagi¹⁵ using spatial-temporal correlational analysis. This phenomenon, characterized by a delay in the response of scalar fluctuations, represents a nonlinear effect that cannot be accounted in the eddy-viscosity/diffusivity-based model. From the conventional derivation of wall asymptotic from either Na and Hanratty¹³ for scalar asymptotic and Leschziner³¹ for velocity asymptotic, an integer exponent in the wall asymptotic is always returned due to the presumption that the scalar fluctuations follow power series expansion of the turbulent velocity fluctuations. However, these discrepancies predominantly occur at small scales, particularly those near the wall.

It should be emphasized that the temporal filtering effect is a physical phenomenon that should be considered when constructing a model of diffusive flux at high Sc numbers. Disregarding the filtering effect can lead to wrong modeling assumptions in the passive scalar model, with all scalar fluctuations respond as readily as velocity perturbations (i.e., The relative proportion of the filtering should be linked to Sc numbers). Although empirical laws, such as the one proposed by Kader,²¹ are widely used, we aim to develop a scalar diffusivity integral-based wall model (i.e., the Pirozzoli¹¹ model) that adheres to the asymptotic behavior of turbulence across a broad range of y⁺ and does not limit its application only to RANS application. Turbulence near the wall, characterized by the friction velocity $u_{τ}$ ⁠, can fluctuate intermittently, leading to variations in Re_τ and, consequently, shifts in the wall scalar flux with respect to y⁺. This variability arises because scalar transport, unlike turbulent flow, cannot be collapsed onto a single universal inner-scaled law. Instead of relying on wall models tied to specific Re_τ, we propose a modification that accounts for transitional behavior in passive scalar transport as Re_τ changes.

In this paper, we focus on advancing the state-of-the-art scalar diffusivity integral law by extending its applicability to high Sc conditions. To address the limitations of the existing scalar diffusivity law, which are often accurate only for low Sc values, we introduce a two-layer approach that separately accounts for the effects of Sc and Re_τ. By deriving relationships from DNS data, we isolate the distinct influences of Sc and Re_τ and integrate them into a single scalar diffusivity-based wall model. The model is also equally applicable for high Prandtl (Pr) heat transfer; however, we will use Sc as the notation throughout the paper and focused on mass transfer applications in the validation case.

II. METHODOLOGY

A. Governing equations

In this work, we are focused in solving the RANS momentum and passive scalar equations. The RANS continuity and momentum equations read

\frac{\partial \bar{u_{j}}}{\partial t} + \bar{u_{i}} \frac{\partial \bar{u_{j}}}{\partial x_{i}} = - \frac{1}{ρ} \frac{\partial \bar{p}}{\partial x_{j}} + ν \frac{\partial^{2} \bar{u_{j}}}{\partial x_{i} \partial x_{i}} - \frac{\partial}{\partial x_{i}} (\bar{u_{i}^{'} u_{j}^{'}}),

(1)

where

\bar{u_{j}}

is the time-averaged velocity in the jth direction,

\bar{p}

is the time-averaged pressure,

ρ

is the fluid density,

ν

is the kinematic viscosity, and

\bar{u_{i}^{'} u_{j}^{'}}

is the Reynolds stress tensor. Similarly, the RANS passive scalar equation reads

\frac{\partial \bar{c}}{\partial t} + \bar{u_{i}} \frac{\partial \bar{c}}{\partial x_{i}} = \frac{\partial}{\partial x_{i}} (α \frac{\partial \bar{c}}{\partial x_{i}}) - \frac{\partial}{\partial x_{i}} (\bar{u_{i}^{'} c^{'}}),

(2)

where

\bar{ϕ}

is the time-averaged concentration of the passive scalar,

α

is the diffusion coefficient of the passive scalar, and

\bar{u_{i}^{'} c^{'}}

is the turbulent scalar flux term. The turbulence scalar flux in (⁠

\bar{u_{i}^{'} c^{'}}

⁠) is modeled using the simple gradient diffusion hypothesis (SGDH) in the current study

\bar{u_{i}^{'} c^{'}} = \frac{ν}{Sc} \frac{\partial \bar{c}}{\partial x_{i}} .

(3)

A review of conventional turbulent scalar flux modeling approaches can be found in the work of Combest et al.³² In this study, we assumed that the SGDH remains applicable to flow mixing away from boundaries or in simulations where the boundary effects are not critical. For the flow accelerated corrosion/erosion project at KTH (interested reader can consult Wong et al.⁵ for further information), we are interested in surface mass transfer process on an attached boundary layer under high Re_τ, where the surface mass flux is the most important quantity in comparison to the fluctuations of scalar away from the wall. This is the reason that SGDH is so prominent in scientific literature with RANS application, for examples, turbulent mass transfer in pipe study by Chen et al.,⁶ high Sc mass transfer for studies after an orifice by representative studies from El-Gammal et al.³³ and Yamagata et al.³⁴ However, employing SGDH can only address the discrepancy in spatial diffusion characteristic of the scalar field, yet overlooks the temporal discrepancies. Therefore, it should acknowledge that wall resolved simulation might still not be able to resolve the near-wall turbulent scalar flux properly due to the lack of consideration of the temporal filtering effects, however, the assessment of such behavior is out of the scope of current study. Instead, the wall model approach is employed to refine the near-wall behavior in high Sc turbulent flows by incorporating a specialized boundary condition for scalar diffusivity. However, the effect of the Sc or Pr can lead to strong anisotropy and nonlinear effects within the flow domain. Therefore, the SGDH or effective diffusivity model as described in Mejia et al.³⁵ may not be valid for the high Sc flows in large eddy simulation (LES). Still it was used in some computational studies for mass transfer process, such as Hartmann et al.³⁶ and Walter et al.³⁷ Mejia et al.³⁵ compares the effective diffusivity model and dynamics calculation of the sub-grid diffusivity approach by Germano et al.³⁸ and found that the dynamic diffusivity model can predict the concentration and scalar fluctuations, while both models correctly predicts the radial distribution of the concentration field in turbulent jet. Therefore, it is advised to use the dynamic diffusivity calculation when conducting high Sc mass transfer process for simulation with LES if scalar flux statistics in bulk are important, such as those will large scale mixing process, or when near wall fluctuation statistics is of importances, such as turbulent mass transfer at relatively low to medium Re_τ.

B. Turbulence modeling

In this work, we applied three turbulence models in validation exercises to evaluate their performance when combining with the two-layer model, the $k - ω$ SST, $k - ϵ$ AKN, and $k - ϵ$ lag Reynolds stress model (RSM) models. The idea is to test both the turbulence model and the two-layer model predictive performance in OpenFOAM.

$k - ω$ SST model: The $k - ω$ SST developed by Menter et al.³⁹ was developed by combining the $k - ω$ and $k - ϵ$ turbulence models. The $k - ω$ formulation is applied in the inner part of the boundary layer, allowing the model to extend seamlessly to the wall through the viscous sublayer. In contrast, the $k - ϵ$ model is used in the free-stream region. The blending of the two models is achieved using a blending function: it equals unity near the wall to prioritize the $k - ω$ model and becomes zero away from the surface, where the $k - ϵ$ model dominates.
$k - ϵ$ AKN model: The $k - ϵ$ AKN model developed by Abe et al.⁴⁰ refines the standard $k - ϵ$ model to improve accuracy in flows with adverse pressure gradients and separation. The AKN model modifies the turbulent viscosity and dissipation rate equations to enhance predictions in the near-wall region. The AKN model introduces damping functions that correct the behavior of turbulent viscosity (⁠ $ν_{t}$ ⁠) and turbulence production terms in the near-wall region

$ν_{t} = C_{μ} \frac{k^{2}}{ϵ} f_{μ},$
(4)

where $f_{μ}$ is a damping function calibrated to reconstruct the behavior of $ν_{t}$ close to the wall. The production term in the $ϵ$ equation is modified to ensure it accurately represents the dissipation rate in these conditions. For details of the modifiation over the standard $k - ϵ$ model, the reader is advised to refer to the original publication by Abe et al.⁴⁰
$k - ϵ$ Lag RSM model: Unlike standard $k - ϵ$ models, which rely on an isotropic eddy viscosity assumption, the Lag RSM by Lardeau and Billard⁴¹ avoids the need of explicitly solving transport equations for individual components of the Reynolds stress tensor $⟨ u_{i}^{'} u_{j}^{'} ⟩$ ⁠, while captruing the anisotropic turbulence effects. The OpenFOAM implementation by Gajetti et al.⁴² is used in the current work.

C. Wall modeling for scalar

When using a coarse grid to model passive scalar transport, the wall scalar flux (⁠

J_{c}

⁠) must be derived based on an equivalence relationship, in accordance with an appropriate wall model or function,

J_{c} = α_{eff} \frac{\partial c}{\partial n} = c_{τ} u_{τ} = \frac{c_{c} - c_{w}}{u_{τ}},

(5)

the effective diffusivity (α_eff) near the wall was determined by utilizing the passive scalar values at both the wall and the first cell center (c_c – c_w), frictional velocity (u_τ), and the near-wall scalar gradient (⁠

\frac{\partial c}{\partial n}

⁠). The turbulent diffusivity (⁠

α_{t}

⁠) is expressed as

α_{t} = α_{eff} - α_{l} = \frac{c_{c} - c_{w}}{c^{+}} u_{τ} {[\frac{\partial c}{\partial n}]}^{- 1} - \frac{ν}{S c} .

(6)

Hence, the

α_{t}

can be derived by subtracting the laminar diffusivity (α_l) from the effective diffusivity (α_eff). The dimensionless scalar (c⁺) two-layer model can be obtained through the wall model (i.e., either the single layer, two layer, or empirical function), based on a local wall y⁺ value. The local wall y⁺ value is estimated based on the local turbulent kinetic energy (⁠

k

⁠),

y^{+} = \frac{y u_{τ}}{ν} = \frac{y u^{*}}{ν} = \frac{y}{ν} C_{μ}^{0.25} k^{0.5},

(7)

where y is the distance of the sampling cell from the wall,

C_{μ}^{0.25}

is taken at 0.09^0.25 as in OpenFOAM's official implementation, the friction velocity (u_τ) can be used to deduce the wall shear stress,

τ_{w} = ρ u_{τ}^{2} .

(8)

III. TWO-LAYER SCALAR DIFFUSIVITY MODEL

A. Need for the consideration of the temporal filtering effects

As noted in the introduction, existing wall models have a limited validity range due to the restrictions in the DNS or experimental data used for their calibration. In addition, the near-wall scalar diffusivity behavior is not respected. Our intended application focuses on Sc number on the order of O(10²) to O(10³), which is beyond the applicability range of the exiting models. For the wall modeling of passive scalar, the scalar diffusivity integral model is adopted in this work due to their relatively good performance and theoretical description of the scalar diffusivity observed in DNS data.

The scalar diffusivity can be directly derived from the mean scalar equation in the inner layer using dimensionless form. An example would be the utilization of a similar eddy viscosity profile as applied by Musker²² in the scalar diffusivity formulation, as performed in Pirrorlzi's work on turbulent pipe flow and Li et al.⁴³ work on non-equilibrium boundary layers. However, when the Sc and Pr are high, the disparity between passive scalar and velocity fields becomes more pronounced. Consequently, the conventional single-layer description for the passive scalar field starts to deviate from the eddy viscosity-based diffusivity model, with their differences scales with Sc or Pr.

Let us revisit the conventional derivation for wall asymptotics, the reader is advised to consult for textbook such as from Leschziner³¹ for more details. Assuming the turbulent fluctuations on a boundary layer flow, u′, v′, w′, can be expressed as power series with arbitrary coefficients (⁠

a_{\dots}, b_{\dots}, c_{\dots}, d_{\dots}),

u^{'} (y^{+}) = a_{1} + a_{2} y^{+} + a_{3} y^{+^{2}} + a_{4} y^{+^{3}} + \dots,

(9)

v^{'} (y^{+}) = b_{1} + b_{2} y^{+} + b_{3} y^{+^{2}} + b_{4} y^{+^{3}} + \dots,

(10)

w^{'} (y^{+}) = c_{1} + c_{2} y^{+} + c_{3} y^{+^{2}} + c_{4} y^{+^{3}} + \dots .

(11)

All turbulent fluctuations (⁠

u^{'}, v^{'}, w'

⁠) must vanish at the wall with

y^{+} = 0

⁠,

a_{1} = b_{2} = b_{3} = 0 .

(12)

The wall-normal gradient for

v'

should vanish at the wall due incompressibility condition,

\frac{\partial u'}{\partial x} = \frac{\partial v'}{\partial y} = \frac{\partial w'}{\partial z} = 0.

(13)

Hence,

v

′ is at least a second-order polynomial without the zero and first order terms,

v^{'} (y^{+}) = b_{3} y^{+^{2}} + b_{4} y^{+^{3}} + \dots .

(14)

To derive the expression for turbulent viscosity, the Reynolds stress reads

{u^{'} v}^{'} (y^{+}) = [a_{1} + a_{2} y^{+} + a_{3} y^{+^{2}} + a_{4} y^{+^{3}} + \dots] [b_{3} y^{+^{2}} + b_{4} y^{+^{3}} + \dots] .

(15)

Rearranging gives

u^{'} v^{'} (y^{+}) = a_{2} b_{3} y^{+^{3}} + {(a}_{2} b_{4} + a_{3} b_{3}) y^{+^{4}} + \dots .

(16)

Therefore,

u^{'} v^{'}

tends to follow

y^{+^{3}}

when

y^{+} \to 0

⁠. For passive scalar, one can express the dimensionless scalar gradient in the form of power series with the arbitary coeffiicents (⁠

d_{\dots}

⁠),

c^{'} (y^{+}) = d_{1} + d_{2} y^{+} + d_{3} y^{+^{2}} + d_{4} y^{+^{3}} + \dots .

(17)

Using the fact of the second derivative of

c'

with respect to

y^{+}

is zero from Na and Hanratty,¹³

c^{'} (y^{+}) = d_{3} y^{+^{2}} + d_{4} y^{+^{3}} + \dots .

(18)

Therefore, it is impossible to deduce a fractional exponent of

u' c'

using this approach due to the equivalent assumptions made in the derivation with velocity and passive scalar. Derivations based on power series implicitly assume that the power exponent must be an integer.

As a result, these derivations lead to the same conclusion as those obtained from wall asymptotic analysis for velocity fluctuations, yielding identical result that assumes all scalar fluctuations arise solely from velocity fluctuations, without accounting for any temporal filtering effects. However, near the wall, scalar diffusivity decreases at a rate more significant than cubic power as in velocity boundary layer. This is due to the additional filtering effect arising from the disparity in the temporal response for the scalar field with high Sc or Pr, which acts on top of the suppression of velocity fluctuations caused by the wall-blocking effect. Temporal filtering arises from the strong disparity in magnitude between molecular viscosity and diffusivity at high Sc or Pr and it is a nonlinear effect (i.e., interaction with temporal fluctuations between velocity and scalar) that cannot be captured without resorting to the DNS data. The effects are more pronounced in regions where diffusive processes dominate, i.e., within the conductive boundary layer. In contrast, the influence of temporal filtering diminishes in regions farther from the wall, where large-scale motions and turbulent mixing prevail, effectively smoothing out the filtering effect.

In this work, we aim to develop a modeling description using a two-layer approach, which signify the need of consideration of filtering effect within the conductive boundary layer where diffusive effect governs, while the region away from the wall can be left untouched due to the turbulence mixing and its smoothing effect of scalar fluctuations.

B. Modeling assumptions

We take the following assumptions when developing the two-layer model description for high Sc and Pr numbers:

The two-layer model is designed to account for temporal filtering effect on the near-wall scalar diffusivity. The temporal filtering effect of wall scalar fluctuations induces a deviation from the typical y⁺³ asymptotic behavior in scalar eddy diffusivity. This deviation is contrary to the expected trend based on analogy with eddy viscosity as mentioned previously in Sec. III A. Therefore, this model is aimed for the very high Sc ranges, where such effect starts to manifest.
The two-layer model assumes, the higher the Sc, the higher the extent of the deviation from the conventional asymptotic scaling (y⁺³ behavior). For Sc at extreme value, the gradient in the diffusivity would be very steep inside the L_cond (due to the reduction of L_cond with Sc). This would give an asymptotic zero wall scalar flux due to complete filtering of the velocity fluctuations.
For passive scalar transport, it is required to consider the outer layer spike for smaller Sc (i.e., Sc > 1) and its effect diminishes for high Sc. We consider the modeling of this as an important aspect for the two-layer model.
Apart from the outer layer spike that predominately occurs at the low Sc ranges, the Reynolds number effect is also required to be considered. Its effect on mean passive scalar profile is relatively strong for Re_τ < 500, while its effect diminishes at higher Re_τ. Transitional behavior due to Re_τ is also a characteristic that the two-layer model is focused on in this work.
Due to the dominant effect of diffusive process in the near wall region, it is assumed that the Sc and Re_τ effects govern each of the layer individually due to the aforementioned reasons. Therefore, it is possible to advance validity of the single layer model to higher Sc numbers. The interdependence between Sc and Re_τ effects is assumed to be negligible in the current model.
The model is developed based on the attached boundary layer (i.e., no non-equilibrium effects such as recirculation) and aimed for very high Sc and Re_τ usage. The validity of the modeling lies within the Sc range where DNS data exists for low Re_τ (i.e., up to 2400).

C. Description of the two-layer model

To consider the effects of variation of Re_τ, Sc, or Pr, we propose to use two distinct layers that can resolve the influence of Sc (or Pr) and Re_τ separately. The two-layer model consists of two regions governed by different values of the model parameter $κ$ _Re and $κ$ _Sc, as shown in Fig. 2. This approach is in analogy with the velocity boundary layer, where a single value of $κ$ (i.e., $κ$ = 0.41) is used to represent the reciprocal of the slope of the velocity boundary layer (i.e., most of the existing model for velocity boundary layer are single layer model, due to the collapse behavior of the boundary layer at high Re_τ). Therefore, the two model parameters $κ$ _Re and $κ$ _Sc are analogous reciprocal values of the slope in each of the regions. As discussed previously, these two regions correspond to the temporal filtering region (which governs the micro-diffusive scalar fluctuations) and the eddy viscosity-controlled region (which governs the scalar mixing due to velocity fluctuations). For areas far from the temporal filtering region, the established principles of turbulent boundary layer dynamics remain applicable without modification (i.e., existing knowledge for high Re_τ and at low Sc can be utilized). The two-layer construction is based on the hypothesis that temporal filtering predominantly occurs within the conductive layer, enabling the development of a fully Sc-dependent model with a dynamic boundary between the two regions. The dynamic boundary moves in accordance with the Sc or Pr. Although the thickness of the conductive layer decreases with increasing Sc, the temporal filtering effect becomes more pronounced due to the increasing temporal response lag, necessitating specialized treatment even under high Sc conditions.

FIG. 2.

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Schematics of the two-layer description of scalar diffusivity with respect to Sc.

The main idea for the separation of the effects of Re_τ and Sc is to allow the deduction of both parametric constants (⁠

κ

_Re and

κ

_Sc) independently using individual set of DNS data without resorting to DNS data defined for both high Sc and high Re_τ, which is currently not available in open literature. This paves a straightforward way to derivation of Sc and Re dependent model using the available DNS data with blending of both effects in the scalar diffusivity function. To develop a two-layer wall model, we modify the single layer scalar diffusivity formulation from Pirozzoli¹¹ so that the temporal filtering behavior in the conductive boundary layer (which is a function of Sc) is considered through a switching function (⁠

β

⁠), while the conventional velocity scaling is applied in the regions within conductive and viscous sub-layer, while keeping other regions intact

α^{+} = \frac{β κ_{Re} y^{+^{3}} + (1 - β) κ_{Sc} y^{+^{γ}}}{{(κ_{Re} y^{+})}^{2} + C_{th}^{2}} .

(19)

Here,

α^{+}

is the dimensionless scalar diffusivity and

γ

is an exponent deviating from the y⁺³ scaling. In the current work,

α = 3.3

⁠, which is close to the value obtained from the Lagrangian simulations in Ref. 12. The normalization constant,

C_{t h}

⁠, is kept as the same in the single-layer model. Such modification of scalar diffusivity preserves the asymptotic behavior of the wall scaling at large y⁺. Hence, this keeps the validity of the model when used in a scale-resolving simulation. The switching function, β, which controls the location of the dynamic boundary, can be calculated from the conductive layer thickness L_cond, as

β = \frac{1}{1 + exp (y^{+} - L_{cond})} .

(20)

The L_cond can be calculated using the correlation derived in Schwertfirm and Manhart⁸ based on DNS data obtained at low Re_τ,

L_{cond} = 11.5 {Sc}^{- 0.29} .

(21)

The correlation in Eq. assumes the dependence of the conductive layer thickness on Sc only. However, comparison of the predictions of the Schwertfirm and Manhart⁸ and the Pirozzoli¹¹ models suggests that the L_cond also depends on Re_τ. A correlation, similar to that in Eq. , was also derived (with the constant and exponent being 11.702 and −0.284, respectively) by Saric et al.⁴⁴ using the dataset from Jayatilleke.²⁰ With the use of Pirozzoli¹¹ data using high Re_τ DNS, we derived the conductive boundary layer thickness correlation as

L_{cond} = 10.5 {Sc}^{- 0.34} .

(22)

In the two-layer model, Eq. is applied for low Re_τ flow conditions, while Eq. is applied for medium to high Re_τ condition.

Figure 3 shows the comparison of Eqs. (21) and (22) to the available DNS and LES data for low and medium to high Re_τ data. Equations (21) or (22) represent the state-of-the-art information on the conductive boundary layer (⁠ $L_{cond}$ ⁠) for low and high Re_τ. While it is not possible to obtain direct comparison in the high Re_τ condition, we chose to compare the $L_{cond}$ low Re_τ DNS data from Schwertfirm and Manhart,⁸ Na and Hanratty;¹³ low Re_τ Hybrid DNS-LES data Hasegawa, and Kasagi;¹⁵ medium Re_τ DNS data from Reeuwijk and Hadžiabdić;⁹ medium Re_τ LES data from Bergent and Tiseji¹⁶ and high Re_τ DNS data from Pirozzoli.¹¹ It can be observed that Eq. (21) has excellent agreement with the low to high Sc under low Re_τ data, while there is some discrepancies for the high Sc under medium Re_τ data. It is expected since the Re_τ condition is lower than the Pirozzoli¹¹ study. However, there is a tendency for the data point approaching Eq. (22) with higher Re_τ condition. Therefore, the choice of the Eq. (21) and Eq. (22) is considered to be satisfactory for the construction of two-layer model. Further validation of the high Re_τ data can be recommended for future work with the availability of the high Sc and high Re_τ DNS data.

FIG. 3.

Comparison of the conductivity layer thickness formula [Eqs. (21) and (22)] with low Reτ DNS data from Schwertfirm and Manhart8 and Na and Hanratty;13 low Reτ Hybrid DNS-LES data Hasegawa, and Kasagi;15 medium Reτ DNS data from Reeuwijk and Hadžiabdić;9 medium Reτ LES data from Bergent and Tiseji;16 and high Reτ DNS data from Pirozzoli.11

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Comparison of the conductivity layer thickness formula [Eqs. (21) and (22)] with low Re_τ DNS data from Schwertfirm and Manhart⁸ and Na and Hanratty;¹³ low Re_τ Hybrid DNS-LES data Hasegawa, and Kasagi;¹⁵ medium Re_τ DNS data from Reeuwijk and Hadžiabdić;⁹ medium Re_τ LES data from Bergent and Tiseji;¹⁶ and high Re_τ DNS data from Pirozzoli.¹¹

In this work, we define the outer layer effect, or outer layer spike, as a phenomenon primarily occurring under low Re and low Sc number conditions. This effect results in an elevated mean scalar value in the logarithmic region of the mean profile, due to the overlapping of the inner and outer layers. The outer layer spike is an effect that distinct the scalar and velocity boundary layer at the logarithmic region for low Re_τ (see Sec. IV A 2 and Fig. 6 for additional description of the effect and the verification of the modeling). To consider the outer layer effect, the dimensionless mean scalar diffusivity equation can be complemented by a term for the outer layer spike (y⁺/ScRe_τ) within the inner scale, as

c^{+} = \int [1 + \frac{y^{+}}{S c {R e}_{τ}}] \frac{S c}{1 + S c [\frac{β κ_{Re} y^{+^{3}} + (1 - β) κ_{Sc} y^{+^{γ}}}{{(κ_{Re} y^{+})}^{2} + C_{th}^{2}}]} d y^{+} .

(23)

The model is then applied directly to the wall modeling subroutine with the input of

y^{+}

⁠, Re_τ and Sc to obtain the desired c⁺ for the correction of the near wall scalar diffusivity as in Eq. . Equation can then be numerically integrated using adaptive Gauss–Kronrod quadrature method to obtain the dimensionless mean scalar profile. The Gauss–Kronrod rule adaptively subdivides the integration interval and applies the integration on each subinterval. It combines a Gaussian quadrature approximation to compute the integral and estimate the error. Subintervals with large errors are further refined until convergence The implementation of adaptive Gauss–Kronrod quadrature method that used in this work was from the Boost C++ package.

D. Description of the functional dependencies for Sc and Re_τ effects

For passive scalar boundary layer,

κ

is not a universal constant but a function of Re_τ and Sc. The strong influence of Re_τ and the outer layer effect on the wall scalar profile results in a poor generalization of the wall model. Here, we develop relationships to include such dependencies into the two-layer model. The dependence of

κ

_Re on Re_τ is deduced in using the open literature DNS database for the cases with Sc = 1 from Kozuka et al.,¹⁷ Pirozzoli,¹⁸ and Pirozzoli et al.¹⁹ The correlation for

κ

_Re is assumed to be in the form of a linear law

κ_{R e} (R e_{τ}) = \{\begin{matrix} a + b R e_{τ} & for R e_{τ} < 500, \\ c + d R e_{τ} & for R e_{τ} \geq 500, \end{matrix}

(24)

where a = 0.3785, b = 1.75 × 10⁻⁴, c = 0.4647, and d = 2.5714 × 10⁻⁶ are the coefficients to obtain the k_Re values. The coefficient

b

and

d

are small, because the transitional behavior in term of the order of magnitude in small; however, they cannot be neglected in terms of the model accuracy since the passive scalar boundary layer does not collapse as in velocity boundary layer when approaching high Re_τ. The validity range of Eq. is for Re_τ in the range between 150 and 4000 and for Sc = 1. The use of the two-region formula in

κ

_Re is due to the sharp transition of the slope for the scalar profile at low Re_τ (especially from 180 to 500 as discussed in Sec. IV A 2). The increasing trend of

κ

_Re with Re_τ is consistent with the results from previous studies in the open literature.

Similarly,

κ

_Sc is deduced using the DNS data from Schwertfirm and Manhart,⁸ Na and Hanratty,¹³ and Hasegawa and Kasagi¹⁵ with various Sc and Re_τ of 150 and 180. The correlation for the k_Sc is assumed to be in the form of a power law

κ_{S c} (S c) = \{\begin{array}{l} 0.41 & for S c < 50, \\ e + f S c^{g} & for S c \geq 50, \end{array}

(25)

where e = 0.425, f = −0.004, and g = 0.5 are coefficients to obtain the

κ

_Sc values. The validity range of Eq. is for Sc in the range between 1 to 2400. The reduction of

κ

_Sc at high Sc is due to the reduction of α⁺ at y⁺ → 0, to fit the increasing gradient within the L_cond, which also decreases with Sc. It is observed that

κ

_Sc remains practically constant across the low Sc range (i.e., Sc < 50), therefore, the constant 0.41 is chosen for convenient. This result is indirectly harmonize with the single layer model from Pirozzoli,¹¹ where the author is able use a single

κ

to correlate the change in Sc behavior at low Sc (from 1 to 16). Therefore, this indirectly shows that the effect of filtering is relatively weak when Sc is small.

In addition to the above-mentioned two region correlations for

κ_{R e}

and

κ_{S c}

⁠, we provide two separate power-law based correlations which prioritizes the accuracy at high Sc rather than Sc close to 1

κ_{R e} (R e_{τ}) = a + b R e_{τ}^{c},

(26)

where a = 0.473, b = −1.145 and c = −0.589. Similarly,

κ

_Sc can be deduced similarly as

κ_{S c} (S c) = d + e S c^{f},

(27)

where d = 0.425, e = −0.004, and f = 0.5. For high Sc conditions in the range of hundreds to thousands, the power-law correlation gives similar predictions as the two-region linear law.

IV. RESULTS AND DISCUSSION

A. Verification of the two-layer wall model

1. Conformance to the scalar diffusivity data

Figure 4 shows the comparison of the dimensionless scalar eddy diffusivity calculated with the two-layer model [Eq. (19)], the single-layer model, and the DNS data for Pr = 16, reported by Pirozzoli in Ref. 11. Although the magnitude of Pr is still relatively small compared to the usual case in FAC on the order of hundreds to thousands, a deviation of the conventional scaling can already be seen. Therefore, it is expected that much stronger deviation can be obtained with increasing Pr or Sc numbers; nevertheless, there is still a significant lack of high Sc DNS data at high Re_τ, in open literature due to the enormous requirement for mesh resolution in high Sc data. The development of an efficient computational approach for high Sc turbulent flow under high Re_τ is recommended for the future work.

FIG. 4.

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Comparison of the dimensionless scalar diffusivity calculated by the new two-layer model, the Pirozzoli model and the DNS data for Pr = 16, reported in Pirozzoli.¹¹

From Fig. 4, it can be seen that the agreement of the two-layer model with the fractional exponent from the DNS data at y⁺ → 0 indicates that the consideration of temporal filtering can be used to improve the prediction for y⁺ < L_cond. We would like to reiterate that such a result is not a direct fit of the Pirozzoli data,¹¹ but rather an indirect combination of both Sc and Re_τ using separate sets of DNS data as discussed in the derivation of functional dependences for $κ_{R e} (R e_{τ})$ and $κ_{S c} (S c)$ ⁠. The two-layer model predicts the DNS data better than the single-layer model in both its original form (with $κ$ = 0.459) due to the two-layer description and this indirectly proves the validity of the two-layer assumption which separate the temporal filtering and eddy-viscosity dominated regions. In contrast, the proposed two-layer model provides a framework to better describe the passive scalar transport, and it has the potential to characterize the Sc variation, while preserving the accuracy of wall model at larger y⁺ which experiences limited effect due to Sc variations.

Figure 5 shows the comparison of the prediction of the dimensionless mean scalar by the single layer, two-layer models and Kader correlation.²¹ The single-layer model can reconstruct the DNS data with the excellent accuracy for this particular case due to the fit of DNS data from the same author. On the contrary, the two-layer model is not a direct fit of the high Pr turbulent pipe flow case, but with a combination of individual data for each layer (i.e., using low $R e_{τ}$ under high Sc for the temporal filtering layer within $L_{cond}$ and eddy-viscosity domainted region using Sc = 1 with various Re_τ). Therefore, the prediction is a combination of the assumption that both effects of $R e_{τ}$ and Sc can be factored out independently. Although the single-layer model for this particular case can be a bit more close to the DNS solution, however, our model is more generally applicable for a wide range of flow cases with different Re_τ and Sc (see Sec. IV A 3). In addition, the error between single and two-layer models lies within 1%, therefore, it is acceptable for such a generally applicable model that defined for high Sc and Pr. However, it should be mentioned that the target application of the two-layer model is aimed at a much higher Sc or Pr ranges than 16 as presented in Fig. 4.

FIG. 5.

Comparison of single layer and two-layer models for DNS data for Pr = 16 under Reτ=1140, reported in Pirozzoli.11

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Comparison of single layer and two-layer models for DNS data for Pr = 16 under Re_τ=1140, reported in Pirozzoli.¹¹

2. Impact of outer layer and Reynolds number effects

Here, we selected a range of DNS data at Sc = 1 from Schwertfirm and Manhart,⁸ Kozuka et al.,¹⁷ Pirozzoli,¹⁸ and Pirozzoli et al.¹⁹ Figure 6 shows that a strong variation in the dimensionless scalar profile is observed for Re_τ values ranging from 180 to 500. However, the influence of Re_τ diminishes as Re_τ increases further. For high Re_τ values (i.e., greater than 500), the scalar fluctuations concentrated close to wall, and this reduces the impact to the scalar fluctuations in the outer layer. Therefore, an inner layer wall model would be sufficed to describe the high Re_τ conditions. In addition, as hinted in the study from Klein et al.,⁴⁵ the extent of outer layer spike is drastically reduced in the case with high Sc or Pr numbers.

FIG. 6.

Outer layer dependence at Sc = 1 with respect to Reτ. DNS data from Schwertfirm and Manhart,8 Kozuka et al.,17 Pirozzoli,18 and Pirozzoli et al.19

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Outer layer dependence at Sc = 1 with respect to Re_τ. DNS data from Schwertfirm and Manhart,⁸ Kozuka et al.,¹⁷ Pirozzoli,¹⁸ and Pirozzoli et al.¹⁹

The outer layer scaling is characterized over a range from 0 to 1, and its influence diminishes at high Sc conditions, where inner layer scaling becomes more pronounced with increasing Sc. Furthermore, the outer layer's length expands as Re increases, leading to a reduced impact of the outer layer at high Re. Therefore, a combination of both Re and Sc effects would produce an even smaller effect on the mean scalar profile. The impact of outer layer spike would be not as pronounced as in the low Sc number. Therefore, it is safe to assume that the outer layer effect is negligible for high Re_τ and high Sc applications. The Sc-dependence on outer layer effect is modeled by an additional Sc in the denominator of the outer layer term in Eq (23).

Figure 7 shows the comparison of the single-layer and two-layer models for the Sc = 1 DNS data under low Re_τ. If the outer layer term is not included in the integration of the scalar diffusivity equation. An under-estimation of the mean scalar can be obtained; therefore, a lower $κ$ value in the logarithmic region is found in open literature (i.e., 0.27) from Schwertfirm and Manhart.⁸ This is because the outer layer spike induces an increase in the mean scalar as compared to the case without outer layer spike. This corresponds a higher slope in the scalar logarithmic layer, and hence a lower $κ$ than 0.41 resulted from the ignorance of such an effect. With the modeling of the outer layer spike as a separated effect, it is possible to deduce the slope as 0.41 for both $κ_{R e}$ and $κ_{S c}$ ⁠, which closely resembles the value defined for the velocity boundary layer. Although $κ_{R e}$ does not usually align with the von Kármán constant in the velocity boundary layer, an agreement is expected to be observed at Sc = 1 under low Re_τ, as the mean velocity and scalar boundary layer profiles are essentially identical for a unity Sc without the outer layer spike. Therefore, the two-layer model can separate the effects due to outer layer and the original effect from the logarithmic region. Figure 8 shows follow-up verification with the Reynolds number effects with the outer layer effect term. The transition in Re_τ is reasonably reproduced by the functional dependencies for $κ_{R e}$ ⁠.

FIG. 7.

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Comparison of single- and two-layer model predictions for Sc = 1 and Re_τ=180.

FIG. 8.

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Reynolds number transition at low and high Re_τ.

3. Verification with data at low to high Sc under low to medium Re_τ

Figures 9(a) and 9(b) show the verification of the two-layer model and the existing wall models against the literature mean profile data for group (a) with Sc = 1–50, and group (b) with Sc = 100–2400. The Jayatilleke P-function²⁰ consistently overpredicts the mean scalar value at high Sc number. The two-layer model demonstrates a good agreement between the model and the low to medium Re_τ DNS and LES datasets with the use of surrogate estimate of the two parametric factors. Although the two-layer wall model assumes that the effect if Sc and Re_τ are independent, it provides a good agreement not only at the low Re_τ but also at medium Re_τ with medium to high Sc range.

FIG. 9.

Comparison between the two-layer wall model and the literature DNS data for low to medium Reτ (scatter data arranged from low to high Reτ). (a) Sc = 1–49 using data from Schwertfirm and Manhart8 and Reeuwijk and Hadžiabdić.9(b) Sc = 100–2400 using data from Na and Hanratty,13 Hasegawa and Kasagi,15 and Bergent and Tiseji.16

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Comparison between the two-layer wall model and the literature DNS data for low to medium Re_τ (scatter data arranged from low to high Re_τ). (a) Sc = 1–49 using data from Schwertfirm and Manhart⁸ and Reeuwijk and Hadžiabdić.⁹(b) Sc = 100–2400 using data from Na and Hanratty,¹³ Hasegawa and Kasagi,¹⁵ and Bergent and Tiseji.¹⁶

Interestingly, our model resembles closely to the prediction from Kader's correlation²¹ at medium Re_τ conditions, without using any of the experimental data used in Kader or any DNS data obtained at high Sc and medium/high Re_τ. Two-layer model provides better agreement at high Sc under medium Re_τ conditions without explicit fitting of the results. In addition, our two-layer model is based on the scalar diffusivity relationship; therefore, the entire wall model shape aligns with the DNS data without abrupt transition as in Kader's correlation. On the other hand, the single layer model from Pirozzoli,¹¹ which is based high Re_τ, provide good agreement with the data at medium Re_τ and low Sc. However, its prediction deviates from the two-layer model and Kader at higher Sc. Since the Reynolds number effect diminishes significantly at high Sc conditions, therefore, such a strong deviation from two-layer model is not justified.

4. Verification with data at low Sc under high Re_τ

A similar verification of the two-layer model for high Re_τ zero pressure gradient thermal boundary layer by Balasubramanian et al.¹⁰ is given in Fig. 10, although the flow is not entirely the same as the boundary layer considered in the model development. The discrepancies between the two-layer model and DNS data (within the inner length scale) in the mean scalar profile between medium to high Re_τ is limited and it diminishes with Sc, even when the transition trend with the Re_τ is not exactly reproduced. The two-layer model predictions closely resemble the case at high Re_τ, with its predictions improves at higher Sc case and higher Re_τ.

FIG. 10.

Comparison between the two-layer wall model and the literature DNS data by Balasubramanian et al.10 for high Reτ (scatter data arranged from low to high Reτ).

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Comparison between the two-layer wall model and the literature DNS data by Balasubramanian et al.¹⁰ for high Re_τ (scatter data arranged from low to high Re_τ).

From all the above-mentioned verification, the two-layer model can provide a better estimate than existing models focusing on high Sc numbers and variable Re_τ conditions.

B. Validation of the two-layer wall model with turbulent pipe mass transfer experiment

The two-layer model is validated by comparing the wall scalar flux predictions against the data from the turbulent mass transfer experiments at high Sc, reported in Shaw and Hanratty⁴⁶ and Berger and Hau.⁴⁷ In Shaw and Hanratty,⁴⁶ an asymptotic correlation for large Sc,

K_{V} = 0.0889 u_{τ} S c^{- 0.704}

(28)

was derived based on theoretical analysis and high Sc experiments [up to O(10⁴)]. K_v in Eq. is the mass transfer coefficient. To convert the correlation, Eq. , to the wall scalar flux, we used the following frictional factor correlation for pipe

f = \frac{1}{{[1.8 log (R e) - 1.5]}^{2}},

(29)

which is used to calculate the frictional velocity,

u_{τ} = \sqrt{\frac{f}{8}} R e,

(30)

for the Shaw–Hanratty correlation. This frictional correlation gives essentially the same friction factor as the highly accuracy implicit Colebrook–White formula. In Berger and Hau,⁴⁷ the high Sc mass transfer pipe experiment was conducted in an electrochemical system. The concentration difference in the pipe flow was measured, and an experimental correlation,

S t = 0.0165 R e^{- 0.14} S c^{- 0.67}

(31)

was derived for the Stanton number (St) as a function of Re and Sc. The validity range for Eq. was reported as Sc between 1000 and 6000 and Re between 8 × 10³ and 2 × 10⁵. These experimental correlations are regarded the validation standard for high Sc simulations.

A turbulent pipe flow with a diameter of D_h with a sufficient inlet entrance length of 20D_h is simulated with a fully developed turbulent velocity profile obtained from the periodic simulation. We studied the performance of the two-layer wall model in simulations with wall refinements at small y⁺ (wall-resolved, WR) and large y⁺ (wall-modeled, WM) values. The required first cell thickness was calculated using the desired y⁺, D_h and the mean velocity (U_m). Depending on the first cell thickness, several tens of prism cells were imposed on the wall. Selected turbulence models with SGDH turbulent scalar flux modeling were used to conduct the simulations. The turbulent Schmidt number (Sc_t) was set to 0.9 due to the small effect of the flow away from boundary layer region in high Sc flows. Figure 11 shows the two level of mesh refinements used in the simulations for both k-ω SST, $k - ϵ$ AKN, and $k - ϵ$ Lag RSM turbulence models. A pre-simulation (Fig. 12) was conducted using a periodic channel with a length of 5D_h to generate a fully developed profile for the turbulence variables. The mapField utility is used to map the outlet patch of the periodic case to the inlet patch of the non-periodic simulation for all the velocity and turbulent variables at the inlet patch.

FIG. 11.

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Two levels of mesh densities from coarse (high Re) to fine (low Re) meshes (from left to right). The near wall spacing is adjusted according to Reynolds number of the simulation.

FIG. 12.

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Periodic (in green) and non-periodic pipe (in grey) simulations.

In the WM simulation, high-Reynolds-number wall functions are applied to all three turbulence models for $k, ω, ϵ, ν_{t}$ ⁠. In contrast, the WR simulation enforces $k, ν_{t}$ are enforced with a fixed-value constraint set to a small value, while hyperbolic tangent blending is used for $ω$ and stepwise blending is applied to $ϵ$ ⁠. The coupling of the WR hydrodynamic simulation with a wall model for $α_{t}$ is justified by the need to accurately capture a highly under-resolved passive scalar field. Under high Sc conditions, accurately capturing the scalar gradient requires using a grid with y⁺ smaller than 0.1, as reported in Nešić et al.⁴⁸ for Sc = 1000. Only at such small y⁺ values the turbulent mass transport become negligible, comparable to molecular diffusive transport. The boundary condition for $α_{t}$ is imposed using the two-layer model.

It is crucial to validate the wall shear predictions from turbulence models before applying the two-layer model. To this end, wall shear stress predictions were compared, showing close agreement with the friction factor correlations across all three turbulence models (Table I). Excellent agreement with friction factor correlation was obtained with the WR k-ω SST model. Nevertheless, WR simulations using k-ϵ models yielded overestimated predictions. Although the near-wall behavior is explicitly modeled with the k-ϵ approach, the near-wall viscosity appears to be overestimated in the region close to the wall. This issue can be partially mitigated by employing the binomial blending option for epsilon at the wall. However, this approach is not entirely accurate, as the blending method does not incorporate a y⁺-dependent selection.

TABLE I.

Comparison of friction factor predictions from different turbulence models.

	Re = 100 000	Re = 200 000
Friction factor correlation	0.0178	0.0155
k-ω SST (Low Re)	0.0178	0.0154
k-ω SST (High Re)	0.0183	0.0153
k-ϵ AKN (Low Re)	0.0194	0.0182
k-ϵ AKN (High Re)	0.0180	0.0152
k-ϵ Lag RSM (Low Re)	0.0236	0.0196
k-ϵ Lag RSM (High Re)	0.0176	0.0149

	Re = 100 000	Re = 200 000
Friction factor correlation	0.0178	0.0155
k-ω SST (Low Re)	0.0178	0.0154
k-ω SST (High Re)	0.0183	0.0153
k-ϵ AKN (Low Re)	0.0194	0.0182
k-ϵ AKN (High Re)	0.0180	0.0152
k-ϵ Lag RSM (Low Re)	0.0236	0.0196
k-ϵ Lag RSM (High Re)	0.0176	0.0149

Figure 13 shows the validation result of the two-layer model in terms of the variation of the Sherwood number (Sh) with respect to Re at a constant Sc of 1000. Small variations are observed with different turbulence models, however, such discrepancies are quite small, i.e., within 3%. The predictive Sh from two-layer model closely aligns with the Berger–Hau and Shaw–Hanratty results for low Re and transitioning to Shaw–Hanratty at high Re numbers. Note that no experimental data employed in deriving the Berger–Hau and Shaw–Hanratty models was used to develop the two-layer model. This is a comparison of a model that is built from bottom-up approach (i.e., numerically integrated from the scalar diffusivity profile) vs experimental data that was obtained independently. The predictions of the single-layer and two-layer model show minimal differences at lower Re values, but the disparity grows with increasing Re.

FIG. 13.

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Comparison of the CFD predictions using the two-layer model, and the correlations from the high Sc mass transfer experiments by Shaw and Hanratty⁴⁶ and Berger and Hau.⁴⁷

Despite this, the relative difference between the two models remains around 15%, primarily attributed to the effect of high-Sc temporal filtering. This observation aligns with Sec. III A, where single-layer model results indicated lower values in the mean scalar profile (and thus, higher wall scalar flux) compared to the Kader correlation at high Sc. The temporal filtering characteristics in our model are evident through its independent alignment with Shaw–Hanratty's results. Conversely, the results from Berger and Hau, derived using heat transfer correlations based on conventional cubic power scaling, show greater discrepancies with increasing Re. In summary, our two-layer adjustment enhances predictions for high-Sc conditions.

V. CONCLUSIONS

The conventional assumption of scalar diffusivity using the analogy with eddy viscosity is not valid for high Sc numbers. Neglecting the temporal filtering effects leads to a non-negligible error (which scale with increasing Sc) in the mean scalar profile and thus, the wall scalar flux at high Sc or Pr conditions. We introduce a two-layer adjustment to model the near-wall scalar diffusivity of in high Sc turbulent flows, with the aims of extending the state-of-the-art scalar diffusivity model to high Sc or Pr conditions. Different from most of the existing empirical models for high Sc conditions, the two-layer model inherit the characteristic of single-layer model, allowing it to be applied as an all y⁺ model in RANS and LES contexts, while being easy to implement.

The findings can be summarized as follows:

The proposed two-layer model separates the effect of Sc and Re_τ and resolves the low pass filtering process that occurs primarily within the conductive layer and leaving the outer layer intact. The model retrains the accuracy at larger $y^{+}$ values based on well-developed information of DNS data at unity Sc at a variety of Re conditions, while absorbing information with Sc dependence using variable Sc at low Re conditions.
The two-layer model yields predictions that agree with the mean scalar profile from the existing DNS and LES data with a wide Sc and Re_τ ranges. This proves that the validity of the hypothesis of mutually exclusive assumption for effect of Sc and Re_τ on mean scalar profile.
The outer layer spike significantly contributes to the discrepancies when comparing the wall model with the DNS data. Ignoring such effects can lead to an underestimation of the $k$ constant for passive scalar at low Re_τ and low Sc conditions.
A complimentary modeling term is used to model the outer layer effect in the two-layer model. The agreement that found with Re-dependent constant in the two-layer model with that defined in velocity boundary layer encourages the belief that the two-layer model is a better alternative in modeling passive scalar transport under variable Sc conditions.
In the turbulent pipe flow test, the two-layer model provides the scalar flux predictions that follow the Berger and Hau and the Shaw and Hanratty correlations, which reduced the percentage error from ∼15% to ∼3%. The agreement of such a bottom-up built wall model and the experimental correlations suggests that the two-layer model adequately captures the relevant filtering effect.

Future work will focus on enhancing the scalar diffusivity relation by incorporating the outer layer effect, which currently limits its effectiveness at high y⁺ conditions. For the current validation test, the objective is to validate the two-layer model in the attached boundary layer to demonstrate its accuracy. However, conventional mass transfer applications in industrial settings typically involve non-equilibrium flows. The utilization of the two-layer scalar diffusivity description to improve the mass transfer process under non-equilibrium conditions, such as those involving roughness features and orifices, is one of the active research area of our project. Additionally, we will conduct wall-modeled large Eddy simulations (LES) of the experimental facility at KTH to estimate the mass transfer rate using the two-layer model and to develop a flow-accelerated corrosion model relevant for lead-cooled fast reactors.

ACKNOWLEDGMENTS

Current research was supported by the Swedish Foundation for Strategic Research (SSF) through Grant No. ARC190043 granted to the Sustainable Nuclear Energy Research in Sweden (SUNRISE) project.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Kin Wing Wong: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Ignas Mickus: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Dmitry Grischenhko: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal). Pavel Kudinov: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request. The two-layer wall model implementation will be shared at https://github.com/kwwong333/two-layer-wall-model.

REFERENCES

1.

A. M.

Hissanaga

,

J. R.

Barbosa

, Jr., and

A. K.

Da Silva

, “

Numerical analysis of inorganic fouling with multi-physics turbulent models

,”

Appl. Therm. Eng.

220

,

119624

(

2023

).

https://doi.org/10.1016/j.applthermaleng.2022.119624

Google Scholar

Crossref

2.

X.

Qiu

,

W.

Chen

,

J.

Yuan

,

X.

Tan

,

S.

Liu

,

G.

Wu

, and

K.

Liu

, “

Equivalent relationship of accelerated corrosion based on the chloride ion diffusion property in calcium sulfoaluminate cement-based pastes

,”

Int. Commun. Heat Mass Transfer

152

,

107283

(

2024

).

https://doi.org/10.1016/j.icheatmasstransfer.2024.107283

Google Scholar

Crossref

3.

Z.

Zhao

,

Z.

Xu

, and

B.

Yu

, “

Numerical study on heat and mass transfer mechanism of dissolved particles in porous media with corroded skeletons

,”

Int. Commun. Heat Mass Transfer

159

,

108158

(

2024

).

https://doi.org/10.1016/j.icheatmasstransfer.2024.108158

Google Scholar

Crossref

4.

Y.

Wang

and

X.

Peng

, “

Numerical study on formation characteristics of particle-crystal mixed fouling in the heat exchanger tube

,”

Int. Commun. Heat Mass Transfer

157

,

107833

(

2024

).

https://doi.org/10.1016/j.icheatmasstransfer.2024.107833

Google Scholar

Crossref

5.

K. W.

Wong

,

I.

Mickus

,

N.

Torkelson

,

S.

Vasudevan

,

H.

Li

,

D.

Grishchenko

, and

P.

Kudinov

, “

Hydrodynamic design of the separate effect test facility for flow-accelerated corrosion and erosion (SEFACE) studies in liquid lead

,”

Nucl. Eng. Des.

417

,

112852

(

2024

).

https://doi.org/10.1016/j.nucengdes.2023.112852

Google Scholar

Crossref

6.

J.

Chen

,

D.

Wang

,

D.

Ewing

, and

C. Y.

Ching

, “

Numerical simulations of mass transfer in turbulent pipe flow at high Schmidt numbers

,”

Heat Mass Transfer

59

,

1333

(

2023

).

https://doi.org/10.1007/s00231-022-03340-w

Google Scholar

Crossref

7.

OECD and Nuclear Energy Agency

,

Handbook on Lead-Bismuth Eutectic Alloy and Lead Properties, Materials Compatibility, Thermalhydraulics and Technologies

(

OECD

,

2015

).

8.

F.

Schwertfirm

and

M.

Manhart

, “

DNS of passive scalar transport in turbulent channel flow at high Schmidt numbers

,”

Int. J. Heat Fluid Flow

28

(

6

),

1204

–

1214

(

2007

).

https://doi.org/10.1016/j.ijheatfluidflow.2007.05.012

Google Scholar

Crossref

9.

M.

van Reeuwijk

and

M.

Hadžiabdić

, “

Modelling high Schmidt number turbulent mass transfer

,”

Int. J. Heat Fluid Flow

51

,

42

–

49

(

2015

).

https://doi.org/10.1016/j.ijheatfluidflow.2014.10.025

Google Scholar

Crossref

10.

A. G.

Balasubramanian

,

L.

Guastoni

,

P.

Schlatter

, and

R.

Vinuesa

, “

Direct numerical simulation of a zero-pressure-gradient turbulent boundary layer with passive scalars up to Prandtl number Pr = 6

,”

J. Fluid Mech.

974

,

A49

(

2023

).

https://doi.org/10.1017/jfm.2023.803

Google Scholar

Crossref

11.

S.

Pirozzoli

, “

An explicit representation for mean profiles and fluxes in forced passive scalar convection

,”

J. Fluid Mech.

968

,

R1

(

2023

).

https://doi.org/10.1017/jfm.2023.591

Google Scholar

Crossref

12.

D. V.

Papavassiliou

and

T. J.

Hanratty

, “

Transport of a passive scalar in a turbulent channel flow

,”

Int. J. Heat Mass Transfer

40

(

6

),

1303

–

1311

(

1997

).

https://doi.org/10.1016/S0017-9310(96)00202-5

Google Scholar

Crossref

13.

Y.

Na

and

T. J.

Hanratty

, “

Limiting behavior of turbulent scalar transport close to a wall

,”

Int. J. Heat Mass Transfer

43

(

10

),

1749

–

1758

(

2000

).

https://doi.org/10.1016/S0017-9310(99)00258-6

Google Scholar

Crossref

14.

Y.

Mito

and

T. J.

Hanratty

, “

Lagrangian stochastic simulation of turbulent dispersion of heat markers in a channel flow

,”

Int. J. Heat Mass Transfer

46

(

6

),

1063

–

1073

(

2003

).

https://doi.org/10.1016/S0017-9310(02)00362-9

Google Scholar

Crossref

15.

Y.

Hasegawa

and

N.

Kasagi

, “

Low-pass filtering effects of viscous sublayer on high Schmidt number mass transfer close to a solid wall

,”

Int. J. Heat Fluid Flow

30

(

3

),

525

–

533

(

2009

).

https://doi.org/10.1016/j.ijheatfluidflow.2009.02.011

Google Scholar

Crossref

16.

R.

Bergant

and

I.

Tiselj

, “

Near-wall passive scalar transport at high Prandtl numbers

,”

Phys. Fluids

19

(

6

),

065105

(

2007

).

https://doi.org/10.1063/1.2739402

Google Scholar

Crossref

17.

M.

Kozuka

,

Y.

Seki

, and

H.

Kawamura

, “

DNS of turbulent heat transfer in a channel flow with a high spatial resolution

,”

Int. J. Heat Fluid Flow

30

(

3

),

514

–

524

(

2009

).

https://doi.org/10.1016/j.ijheatfluidflow.2009.02.023

Google Scholar

Crossref

18.

S.

Pirozzoli

, “

Prandtl number effects on passive scalars in turbulent pipe flow

,”

J. Fluid Mech.

965

,

A7

(

2023

).

https://doi.org/10.1017/jfm.2023.387

Google Scholar

Crossref

19.

S.

Pirozzoli

,

M.

Bernardini

, and

P.

Orlandi

, “

Passive scalars in turbulent channel flow at high Reynolds number

,”

J. Fluid Mech.

788

,

614

–

639

(

2016

).

https://doi.org/10.1017/jfm.2015.711

Google Scholar

Crossref

20.

C. L. V.

Jayatilleke

, “

The influence of Prandtl number and surface roughness on the resistance of the laminar sub-layer to momentum and heat transfer

,” Ph.D. thesis (

Imperial College London

,

1966

).

Google Scholar

21.

B. A.

Kader

, “

Temperature and concentration profiles in fully turbulent boundary layers

,”

Int. J. Heat Mass Transfer

24

(

9

),

1541

–

1544

(

1981

).

https://doi.org/10.1016/0017-9310(81)90220-9

Google Scholar

Crossref

22.

A. J.

Musker

, “

Explicit expression for the smooth wall velocity distribution in a turbulent boundary layer

,”

AIAA J.

17

(

6

),

655

–

657

(

1979

).

https://doi.org/10.2514/3.61193

Google Scholar

Crossref

23.

Y.

Gao

,

M.

Takahashi

, and

M.

Nomura

, “

Characteristics of iron and nickel diffusion in molten lead-bismuth eutectic

,”

Mech. Eng. J.

2

(

6

),

15

-

00149

(

2015

).

https://doi.org/10.1299/mej.15-00149

Google Scholar

Crossref

24.

Y.

Gao

,

M.

Takahashi

,

C.

Cavallotti

, and

G.

Raos

, “

Molecular dynamics simulation of metallic impurity diffusion in liquid lead-bismuth eutectic (LBE)

,”

J. Nucl. Mater.

501

,

253

–

260

(

2018

).

https://doi.org/10.1016/j.jnucmat.2018.01.044

Google Scholar

Crossref

25.

K.

Suga

, “

Computation of high Prandtl number turbulent thermal fields by the analytical wall-function

,”

Int. J. Heat Mass Transfer

50

(

25–26

),

4967

–

4974

(

2007

).

https://doi.org/10.1016/j.ijheatmasstransfer.2007.09.020

Google Scholar

Crossref

26.

B. M.

Mitrovic

,

P. M.

Le

, and

D. V.

Papavassiliou

, “

On the Prandtl or Schmidt number dependence of the turbulent heat or mass transfer coefficient

,”

Chem. Eng. Sci.

59

(

3

),

543

–

555

(

2004

).

https://doi.org/10.1016/j.ces.2003.09.039

Google Scholar

Crossref

27.

I.

Calmet

and

J.

Magnaudet

, “

Large-eddy simulation of high-Schmidt number mass transfer in a turbulent channel flow

,”

Phys. Fluids

9

(

2

),

438

–

455

(

1997

).

https://doi.org/10.1063/1.869138

Google Scholar

Crossref

28.

Y.-H.

Dong

,

X.-Y.

Lu

, and

L.-X.

Zhuang

, “

Large eddy simulation of turbulent channel flow with mass transfer at high-Schmidt numbers

,”

Int. J. Heat Mass Transfer

46

(

9

),

1529

–

1539

(

2003

).

https://doi.org/10.1016/S0017-9310(02)00456-8

Google Scholar

Crossref

29.

K. W.

Wong

,

L.

Bures

, and

K.

Mikityuk

, “

Interface tracking investigation of the sliding bubbles effects on heat transfer in the laminar regime

,”

Nucl. Technol.

208

(

8

),

1266

–

1278

(

2022

).

https://doi.org/10.1080/00295450.2021.1971025

Google Scholar

Crossref

30.

H.

Tennekes

and

J. L.

Lumley

,

A First Course in Turbulence

(

MIT Press

,

1972

).

Google Scholar

Crossref

31.

M.

Leschziner

,

Statistical Turbulence Modelling for Fluid Dynamics-Demystified: An Introductory Text for Graduate Engineering Students

(

World Scientific

,

2015

).

Google Scholar

Crossref

32.

D. P.

Combest

,

P. A.

Ramachandran

, and

M. P.

Dudukovic

, “

On the gradient diffusion hypothesis and passive scalar transport in turbulent flows

,”

Ind. Eng. Chem. Res.

50

(

15

),

8817

–

8823

(

2011

).

https://doi.org/10.1021/ie200055s

Google Scholar

Crossref

33.

M.

El-Gammal

,

W. H.

Ahmed

, and

C. Y.

Ching

, “

Investigation of wall mass transfer characteristics downstream of an orifice

,”

Nucl. Eng. Des.

242

,

353

–

360

(

2012

).

https://doi.org/10.1016/j.nucengdes.2011.10.009

Google Scholar

Crossref

34.

T.

Yamagata

,

A.

Ito

,

Y.

Sato

, and

N.

Fujisawa

, “

Experimental and numerical studies on mass transfer characteristics behind an orifice in a circular pipe for application to pipe-wall thinning

,”

Exp. Therm. Fluid Sci.

52

,

239

–

247

(

2014

).

https://doi.org/10.1016/j.expthermflusci.2013.09.017

Google Scholar

Crossref

35.

J. M.

Mejía

,

A.

Sadiki

,

A.

Molina

,

F.

Chejne

, and

P.

Pantangi

, “

Large eddy simulation of the mixing of a passive scalar in a high-Schmidt turbulent jet

,”

J. Fluids Eng.

137

(

3

),

031301

(

2015

).

https://doi.org/10.1115/1.4029224

Google Scholar

Crossref

36.

H.

Hartmann

,

J.

Derksen

,

C.

Montavon

,

J.

Pearson

,

I.

Hamill

, and

H.

Van den Akker

, “

Assessment of large eddy and RANS stirred tank simulations by means of LDA

,”

Chem. Eng. Sci.

59

(

12

),

2419

–

2432

(

2004

).

https://doi.org/10.1016/j.ces.2004.01.065

Google Scholar

Crossref

37.

M.

Walter

,

N.

Kornev

,

V. L.

Zhdanov

, and

E. P.

Hassel

, “

Turbulent mixing with chemical reaction in a coaxial jet mixer

,” in

Proceedings of the Sixth International Symposium On Turbulence, Heat and Mass Transfer

(

Begel House Inc.

,

2009

).

Google Scholar

Crossref

38.

M.

Germano

,

U.

Piomelli

,

P.

Moin

, and

W. H.

Cabot

, “

A dynamic subgrid-scale eddy viscosity model

,”

Phys. Fluids Fluid Dyn

3

(

7

),

1760

–

1765

(

1991

).

https://doi.org/10.1063/1.857955

Google Scholar

Crossref

39.

F. R.

Menter

,

M.

Kuntz

, and

R.

Langtry

, “

Ten years of industrial experience with the SST turbulence model

,”

Turbul., Heat Mass Transfer

4

(

1

),

625

–

632

(

2003

).

Google Scholar

40.

K.

Abe

,

T.

Kondoh

, and

Y.

Nagano

, “

A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows—II. Thermal field calculations

,”

Int. J. Heat Mass Transfer

38

(

8

),

1467

–

1481

(

1995

).

https://doi.org/10.1016/0017-9310(94)00252-Q

Google Scholar

Crossref

41.

S.

Lardeau

and

F.

Billard

, “

Development of an elliptic-blending lag model for industrial applications

,” AIAA Paper No. 2016-1600,

2016

.

Google Scholar

42.

E.

Gajetti

,

L.

Marocco

,

G.

Boccardo

,

A.

Buffo

, and

L.

Savoldi

, “

Implementation of a lag elliptic-blending model for RANS equations in OpenFOAM

,”

OpenFOAM J.

4

,

79

–

91

(

2024

).

https://doi.org/10.51560/ofj.v4.133

Google Scholar

Crossref

43.

Y.

Li

,

F.

Ries

,

W.

Leudesdorff

,

K.

Nishad

,

A.

Pati

,

C.

Hasse

,

J.

Janicka

,

S.

Jakirlić

, and

A.

Sadiki

, “

Non-equilibrium wall functions for large Eddy simulations of complex turbulent flows and heat transfer

,”

Int. J. Heat Fluid Flow

88

,

108758

(

2021

).

https://doi.org/10.1016/j.ijheatfluidflow.2020.108758

Google Scholar

Crossref

44.

S.

Saric

,

A.

Ennemoser

,

B.

Basara

,

H.

Petutschnig

,

C.

Irrenfried

,

H.

Steiner

, and

G.

Brenn

, “

Improved modeling of near-wall heat transport for cooling of electric and hybrid powertrain components by high Prandtl number flow

,”

SAE Int. J. Engines

10

(

3

),

778

–

784

(

2017

).

https://doi.org/10.4271/2017-01-0621

Google Scholar

Crossref

45.

M.

Klein

,

H.

Schmidt

, and

D. O.

Lignell

, “

Stochastic modeling of surface scalar-flux fluctuations in turbulent channel flow using one-dimensional turbulence

,”

Int. J. Heat Fluid Flow

93

,

108889

(

2022

).

https://doi.org/10.1016/j.ijheatfluidflow.2021.108889

Google Scholar

Crossref

46.

D. A.

Shaw

and

T. J.

Hanratty

, “

Influence of Schmidt number on the fluctuations of turbulent mass transfer to a wall

,”

AIChE J.

23

(

2

),

160

–

169

(

1977

).

https://doi.org/10.1002/aic.690230204

Google Scholar

Crossref

47.

F.

Berger

and

K.-F.-L.

Hau

, “

Mass transfer in turbulent pipe flow measured by the electrochemical method

,”

Int. J. Heat Mass Transfer

20

(

11

),

1185

–

1194

(

1977

).

https://doi.org/10.1016/0017-9310(77)90127-2

Google Scholar

Crossref

48.

S.

Nešić

,

J.

Postlethwaite

, and

D.

Bergstrom

, “

Calculation of wall-mass transfer rates in separated aqueous flow using a low Reynolds number κ-ε model

,”

Int. J. Heat Mass Transfer

35

(

8

),

1977

–

1985

(

1992

).

https://doi.org/10.1016/0017-9310(92)90200-C

Google Scholar

Crossref

This work is licensed under a Creative Commons Attribution 4.0 International License .

A modified two-layer scalar diffusivity description for high Schmidt and Prandtl turbulent boundary layers

NOMENCLATURE

Acronyms

I. INTRODUCTION

II. METHODOLOGY

A. Governing equations

B. Turbulence modeling

C. Wall modeling for scalar

III. TWO-LAYER SCALAR DIFFUSIVITY MODEL

A. Need for the consideration of the temporal filtering effects

B. Modeling assumptions

C. Description of the two-layer model

D. Description of the functional dependencies for Sc and Re_τ effects

IV. RESULTS AND DISCUSSION

A. Verification of the two-layer wall model

1. Conformance to the scalar diffusivity data

2. Impact of outer layer and Reynolds number effects

3. Verification with data at low to high Sc under low to medium Re_τ

4. Verification with data at low Sc under high Re_τ

B. Validation of the two-layer wall model with turbulent pipe mass transfer experiment

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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A modified two-layer scalar diffusivity description for high Schmidt and Prandtl turbulent boundary layers

NOMENCLATURE

Acronyms

I. INTRODUCTION

II. METHODOLOGY

A. Governing equations

B. Turbulence modeling

C. Wall modeling for scalar

III. TWO-LAYER SCALAR DIFFUSIVITY MODEL

A. Need for the consideration of the temporal filtering effects

B. Modeling assumptions

C. Description of the two-layer model

D. Description of the functional dependencies for Sc and Reτ effects

IV. RESULTS AND DISCUSSION

A. Verification of the two-layer wall model

1. Conformance to the scalar diffusivity data

2. Impact of outer layer and Reynolds number effects

3. Verification with data at low to high Sc under low to medium Reτ

4. Verification with data at low Sc under high Reτ

B. Validation of the two-layer wall model with turbulent pipe mass transfer experiment

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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D. Description of the functional dependencies for Sc and Re_τ effects

3. Verification with data at low to high Sc under low to medium Re_τ

4. Verification with data at low Sc under high Re_τ