# Advances in Frazil Ice Evolution Mechanisms and Numerical Modelling in Rivers and Channels in Cold Regions

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## Abstract

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## 1. Introduction

## 2. Laboratory Studies and Field Observation

#### 2.1. Laboratory Studies

#### 2.1.1. Experimental Facilities

#### 2.1.2. Observation Instruments

#### 2.2. Field Observation

## 3. Heat Exchange and Supercooling Processes

#### 3.1. Heat Exchange

#### 3.1.1. Water–Atmosphere Heat Exchange

_{wa}is assumed to be essentially the same everywhere at the same latitude for the selection of parameters in this method, and 20 W/(m

^{2}·°C) is typically taken as the value for rivers in the United States and at high latitudes in northern China [66]. The heat exchange coefficient h

_{wa}is derived from numerous observed experimental datasets and is not precise enough for short-term forecasting, despite the generalized linear algorithm being easy and quick to calculate. Hence, accurate calculations of the water–atmosphere heat flux that use processes such as atmospheric radiation and water–air convective heat transfer are still required when conditions permit it. Richard [67] also presents heat budget and heat transfer equations (Table 1, No. 3). Like Shen’s model, Richard’s heat transfer model also considers long-wave radiation, short-wave radiation, water–-air convection heat transfer, and evaporative condensation heat flux, but Richard also takes into account the effect of precipitation on the water temperature, and provides a formula for heat fluxes due to precipitation.

#### 3.1.2. Water–Boundary Heat Exchange

^{−2}on the beginning of every month between November and April. Evans et al. [58] calculated it to be about 5 W·m

^{−2}in November. This indicates that the heat flux between the water and the boundary (river channels, etc.) is small and the heat exchange between water and external sources mainly comprises water–atmosphere exchanges [7,68].

#### 3.1.3. Water–Ice Heat Exchange

_{u}= 1, only conduction occurs. The amount of convective heat transfer between water and frazil ice is judged by the size of the ice particles and the smallest eddies present in the bulk flow [70]. If the particle size is smaller than the smallest existing eddies, N

_{u}increases to a little above 1. When a crystal is larger than the existing eddies, the value of N

_{u}will be considerably greater than 1. In the formula for calculating h

_{wi}, l represents the characteristic length of particles, but researchers have sometimes taken a different approach to the value. Daly [12] parametrized the N

_{u}values of the large and small size particles based on experiments and theory, using the ratio of the ice particle radius to the minimum vortex scale, m*, as the criterion for judging the particles to be large or not (Table 1, No. 7). The parameter N

_{uT}reveals the relative heat transfer coefficient for each disc radius at a common length scale. The parameters N

_{u}and N

_{uT}are easily confused [71,72,73] and need to be noted in the calculations. The current water–ice heat exchange model is well established and can be applied to all water–frazil ice heat exchange calculations with no limitations.

Model | No. | Year | Researcher | Formula | Remark |
---|---|---|---|---|---|

Heat transfer between water and air | 1 | 1984 | Shen [7] | $\sum {\varphi}_{wa}={\varphi}_{s}+{\varphi}_{b}+{\varphi}_{e}+{\varphi}_{c}$ ${\varphi}_{s}=\left(1-\alpha \right){\varphi}_{ri}$ ${\varphi}_{ri}=\left[a-b\left(\varphi -50\right)\right]\left(1-0.0065{C}^{2}\right)$ ${\varphi}_{b}={\varphi}_{bs}-{\varphi}_{bn}=1.1358\times {10}^{-7}\left[{T}_{sk}^{4}-\left(1+{k}_{C}{C}^{2}\right)\xb7\left(c+d\sqrt{{e}_{a}}\right){T}_{ak}^{4}\right]$ ${\varphi}_{e}={C}_{e}\left(1.56{K}_{n}+6.08{V}_{a}\right)\left({e}_{s}-{e}_{a}\right)$ ${K}_{n}=8.0+0.35\left({T}_{s}-{T}_{a}\right)$ ${\varphi}_{c}={C}_{c}\left({K}_{n}+3.9{V}_{a}\right)\left({T}_{s}-{T}_{a}\right)$ | ${\varphi}_{wa}$ is the heat transfer between water and air; ${\varphi}_{s}$ is short-wave radiation, cal·cm^{−2}·day^{−1} ${\varphi}_{b}$$\mathrm{is}\mathrm{long}\text{-}\mathrm{wave}\mathrm{radiation},\mathrm{cal}\xb7{\mathrm{cm}}^{-2}\xb7{\mathrm{day}}^{-1};{\varphi}_{e}$$\mathrm{is}\mathrm{the}\mathrm{evapo}\text{-}\mathrm{condensation}\mathrm{flux},\mathrm{cal}\xb7{\mathrm{cm}}^{-2}\xb7{\mathrm{day}}^{-1};{\varphi}_{c}$ is the conductive heat transfer, cal·cm^{−2}·day^{−1};$\alpha $$\mathrm{is}\mathrm{a}\mathrm{coefficient}\mathrm{approximately}\mathrm{equal}\mathrm{to}0.1\mathrm{for}\mathrm{the}\mathrm{water}\u2019\mathrm{s}\mathrm{surface};{\varphi}_{ri}$ is the incoming short-wave radiation, cal·cm ^{−2}·day^{−1};$\varphi $$\mathrm{is}\mathrm{the}\mathrm{latitude}\mathrm{in}\mathrm{degrees};a,b$$\mathrm{are}\mathrm{the}\mathrm{constants}\mathrm{that}\mathrm{represent}\mathrm{variations}\mathrm{in}\mathrm{the}\mathrm{solar}\mathrm{radiation}\mathrm{under}\mathrm{a}\mathrm{clear}\mathrm{sky};C$ is the cloud cover in tenths; ${\varphi}_{bs}$$\mathrm{is}\mathrm{the}\mathrm{long}\text{-}\mathrm{wave}\mathrm{radiation}\mathrm{emitted}\mathrm{from}\mathrm{the}\mathrm{water}\mathrm{surface}\mathrm{or}\mathrm{the}\mathrm{ice}\mathrm{cover},\mathrm{cal}\xb7{\mathrm{cm}}^{-2}\xb7{\mathrm{day}}^{-1};{\varphi}_{bn}$$\mathrm{is}\mathrm{the}\mathrm{net}\mathrm{atmospheric}\mathrm{thermal}\mathrm{radiation}\mathrm{absorbed}\mathrm{by}\mathrm{the}\mathrm{water}\mathrm{bod},\mathrm{cal}\xb7{\mathrm{cm}}^{-2}\xb7{\mathrm{day}}^{-1};{T}_{sk}$$\mathrm{is}\mathrm{the}\mathrm{water}\mathrm{or}\mathrm{ice}\mathrm{surface}\mathrm{temperature}\mathrm{in}\mathrm{degrees}\mathrm{Kelvin};{T}_{ak}$$\mathrm{is}\mathrm{the}\mathrm{air}\mathrm{temperature}\mathrm{in}\mathrm{degrees}\mathrm{Kelvin};{e}_{a}$$\mathrm{is}\mathrm{the}\mathrm{vapor}\mathrm{pressure}\mathrm{of}\mathrm{air}\mathrm{at}\mathrm{the}\mathrm{temperature}{T}_{ak}$$,\mathrm{mb};{k}_{C}$ = 0.0017; c = 0.55; d = 0.052; ${C}_{e}$$\mathrm{is}\mathrm{a}\mathrm{coefficient}\mathrm{when}\mathrm{the}\mathrm{river}\mathrm{surface}\mathrm{is}\mathrm{covered}\mathrm{by}\mathrm{ice};{K}_{n}$$\mathrm{is}\mathrm{a}\mathrm{coefficient}\mathrm{that}\mathrm{accounts}\mathrm{for}\mathrm{the}\mathrm{effort}\mathrm{of}\mathrm{free}\mathrm{convection};{V}_{a}$$\mathrm{is}\mathrm{the}\mathrm{wind}\mathrm{velocity}\mathrm{at}2\mathrm{m}\mathrm{above}\mathrm{the}\mathrm{water}\u2019\mathrm{s}\mathrm{surface},\mathrm{m}/\mathrm{s};{e}_{s}$$\mathrm{is}\mathrm{the}\mathrm{saturated}\mathrm{vapor}\mathrm{pressure}\mathrm{at}\mathrm{temperature}{T}_{s}$, mb; ${T}_{s},{T}_{a}$$\mathrm{are}\mathrm{the}\mathrm{river}\mathrm{surface}\mathrm{temperature}\mathrm{and}\mathrm{air}\mathrm{temperature}\mathrm{at}2\mathrm{m}\mathrm{above}\mathrm{the}\mathrm{water}\mathrm{surface},\mathrm{in}\mathrm{degrees}\mathrm{Celsius};{C}_{c}$ is a coefficient, similar to the evapo-condensation flux; |

2 | 1991 | Lai [74] | ${Q}_{wa}={h}_{wa}\left({T}_{w}-{T}_{a}\right)$ | ${h}_{wa}$$\mathrm{is}\mathrm{the}\mathrm{heat}\mathrm{exchange}\mathrm{coefficient}\mathrm{between}\mathrm{water}\mathrm{and}\mathrm{air},\mathrm{which}\mathrm{is}\mathrm{about}20\mathrm{W}\xb7{\mathrm{m}}^{-2}\xb7$°C^{−1}$;{T}_{w}$$\mathrm{is}\mathrm{the}\mathrm{water}\mathrm{temperature},$ °C; ${T}_{a}$$\mathrm{is}\mathrm{the}\mathrm{air}\mathrm{temperature},$ °C; | |

3 | 2015 | Richard [67] | ${Q}_{wa}=\left({Q}_{l{w}_{\downarrow}}+{Q}_{l{w}_{\uparrow}}\right)+{Q}_{s{w}_{\downarrow}}+{Q}_{L}+{Q}_{S}+{Q}_{P}$ ${Q}_{l{w}_{\uparrow}}={\epsilon}_{w}\sigma {\left({T}_{w}+273.15\right)}^{4}$ ${Q}_{l{w}_{\downarrow}}=\left(1-{\alpha}_{w}^{LW}\right){\epsilon}_{a}\sigma {\left({T}_{air}+273.15\right)}^{4}$ ${Q}_{s{w}_{\downarrow}}=\left(1-{\alpha}_{w}^{SW}\right){Q}_{S{W}_{\downarrow}}^{global}$ $\tau ={\rho}_{air}{C}_{Dr}{S}_{r}{}^{2}$ ${Q}_{S}={\rho}_{air}{c}_{pa}{C}_{Hr}{S}_{r}\left({T}_{S}-{T}_{r}\right)$ ${Q}_{L}={\rho}_{air}{L}_{v}{C}_{Er}{S}_{r}\left({q}_{S}-{q}_{r}\right)$ ${Q}_{P}={I}_{S}\left[{L}_{i}+{c}_{pi}\left({T}_{w}-{T}_{a}\right)\right]$ | ${Q}_{wa}$$\mathrm{is}\mathrm{the}\mathrm{heat}\mathrm{transfer}\mathrm{between}\mathrm{water}\mathrm{and}\mathrm{air};{Q}_{l{w}_{\downarrow}}+{Q}_{l{w}_{\uparrow}}$$\mathrm{is}\mathrm{net}\mathrm{long}\text{-}\mathrm{wave}\mathrm{radiation};{Q}_{s{w}_{\downarrow}}$$\mathrm{is}\mathrm{net}\mathrm{short}\text{-}\mathrm{wave}\mathrm{radiation};{Q}_{L}+{Q}_{S}$$\mathrm{are}\mathrm{turbulent}\mathrm{surface}\mathrm{fluxes};{Q}_{P}$$\mathrm{is}\mathrm{precipitations};\left[{}_{\uparrow}\right]$$\mathrm{and}\left[{}_{\downarrow}\right]$$\mathrm{are}\mathrm{used}\mathrm{to}\mathrm{describe}\mathrm{the}\mathrm{direction}\mathrm{of}\mathrm{the}\mathrm{heat}\mathrm{flux}\mathrm{relative}\mathrm{to}\mathrm{the}\mathrm{water}\mathrm{body};\left[{}_{\uparrow}\right]$ means the water experiences a heat loss or negative heat flux, and $\left[{}_{\downarrow}\right]$ means the opposite; ${\epsilon}_{w}$$\mathrm{is}\mathrm{the}\mathrm{water}\mathrm{surface}\mathrm{emissivity},{\epsilon}_{w}\approx 0.97$$;$$\mathrm{is}\mathrm{the}\mathrm{Stefan}\u2013\mathrm{Boltzmann}\mathrm{constant},\sigma =5.670\times {10}^{-8}{\mathrm{W}\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$$;{T}_{air}$ is the air temperature, °C; ${\alpha}_{w}^{LW}$ is the water albedo for long-wave radiation (approximately equal to 0.03); ${Q}_{S{W}_{\downarrow}}^{global}$$\mathrm{is}\mathrm{the}\mathrm{global}\mathrm{incoming}\mathrm{short}\text{-}\mathrm{wave}(\mathrm{or}\mathrm{solar})\mathrm{radiation};{\alpha}_{w}^{SW}$ is the short-wave water surface albedo; $\tau $$\mathrm{is}\mathrm{the}\mathrm{momentum}\mathrm{used}\mathrm{to}\mathrm{compute}\mathrm{the}\mathrm{latent}\mathrm{and}\mathrm{sensible}\mathrm{heat}\mathrm{surface}\mathrm{turbulent}\mathrm{fluxes}\mathrm{by}\mathrm{a}\mathrm{bulk}\mathrm{flux}\mathrm{algorithm};{\rho}_{air}$ is the air density; ${S}_{r}$ is the effective wind speed at a reference height; ${C}_{Dr}$, ${C}_{Hr}$, and ${C}_{Er}$, are the drag coefficients and heat transfer coefficients for sensible and latent heat, respectively ${c}_{pa}$ is the specific heat of air; ${L}_{v}$ is the latent heat of vaporization (or sublimation); ${c}_{pi}$$\mathrm{is}\mathrm{the}\mathrm{specific}\mathrm{heat}\mathrm{of}\mathrm{ice};{I}_{S}$ is the rate of falling snow mass per unit area; ${T}_{S}$ and ${T}_{r}$ are the temperature at the surface and the potential temperature (the temperature that a parcel of fluid would acquire if adiabatically brought to a standard reference pressure) at a reference height, respectively; ${q}_{S}$$\mathrm{and}{q}_{r}$ are the humidity at the surface and the potential specific humidity at a reference height, respectively; | |

Heat transfer between water and frazil ice | 4 | 1984 | Omstedt [75] | ${Q}_{wi}={h}_{wi}\left({T}_{i}-{T}_{w}\right)$ ${h}_{wi}=\frac{{N}_{u}{k}_{w}}{l}$ | ${h}_{wi}$ is the coefficient of the heat exchange between water and ice; ${N}_{u}$ is the Nusselt number, $\mathrm{which}\mathrm{is}\mathrm{used}\mathrm{as}\mathrm{a}\mathrm{constant}\mathrm{in}\mathrm{this}\mathrm{article},{N}_{u}=6.0$$;{k}_{w}$$\mathrm{is}\mathrm{the}\mathrm{thermal}\mathrm{conductivity};l$$\mathrm{represents}\mathrm{the}\mathrm{characteristic}\mathrm{length}\mathrm{of}\mathrm{ice};l$$\mathrm{is}2{R}_{i}$$\mathrm{in}\mathrm{this}\mathrm{article};{R}_{i}$ is the radius of the spherical ice particles, i = 1, 2…. |

5 | 1985 | Omstedt [76] | ${N}_{u}$$\mathrm{is}\mathrm{used}\mathrm{as}\mathrm{a}\mathrm{constant}\mathrm{in}\mathrm{this}\mathrm{article},\mathrm{and}\mathrm{ice}\mathrm{crystal}\mathrm{particles}\mathrm{are}\mathrm{assumed}\mathrm{to}\mathrm{be}\mathrm{disc}\text{-}\mathrm{shaped}\mathrm{in}\mathrm{this}\mathrm{article};l$ is the thickness of the ice crystal; | ||

6 | 1994 | Svensson [77] | |||

7 | 1984 | Daly [12] | ${N}_{uT}=\left(\frac{1}{{m}^{*}}\right)+0.17{P}_{r}^{0.5};$ $if{m}^{*}<\frac{1}{{P}_{r}^{0.5}}$ ${N}_{uT}=\left(\frac{1}{{m}^{*}}\right)+0.55{\left(\frac{{P}_{r}}{{m}^{*}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.};$ $if\frac{1}{{P}_{r}^{0.5}}<{m}^{*}<1$ For large particles, i.e., m* > 1: ${N}_{uT}=1.1\left[\left(\frac{1}{{m}^{*}}\right)+0.8{\alpha}_{T}{}^{0.035}{\left(\frac{{P}_{r}}{{m}^{*}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right];$ $if{\alpha}_{T}{m}^{*}{}^{4/3}\le 1000$ ${N}_{uT}=1.1\left[\left(\frac{1}{{m}^{*}}\right)+0.8{\alpha}_{T}{}^{0.24}{\left({P}_{r}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right];$ $if{\alpha}_{T}{m}^{*}{}^{4/3}>1000$ ${m}^{*}=\frac{r}{\eta}$ ${\alpha}_{T}=\sqrt{2k/U}$ | $\mathrm{Ice}\mathrm{crystal}\mathrm{particles}\mathrm{are}\mathrm{also}\mathrm{assumed}\mathrm{to}\mathrm{be}\mathrm{disc}\text{-}\mathrm{shaped},l={\left({A}_{s}/4\pi \right)}^{0.5}$; ${N}_{uT}$ is the “turbulent” Nusselt number; ${m}^{*}$$\mathrm{is}\mathrm{the}\mathrm{ratio}\mathrm{between}\mathrm{the}\mathrm{face}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{ice}\mathrm{crystal}\mathrm{and}\mathrm{the}\mathrm{Kolmogorov}\mathrm{length}\mathrm{scale},\eta ={\left({\gamma}^{3}/\epsilon \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ ${P}_{r}$ is the Prandtl Number; $k$ is the turbulence intensity; U is the mean flow velocity. | |

8 | 2007 | Holland [69] | ${N}_{u}={N}_{uT}\times {m}^{*}$ | A correction to “Frazil evolution in channels” by Lars Hammar and Hung-Tao Shen [70]. |

#### 3.2. Supercooling Process

_{min}to a stable supercooling temperature Tr when the rate of the latent heat release from the ice in the water exceeds the rate of heat loss from the water body to the outside. According to laboratory research [8], there is a strong correlation between the water supercooling process and the external temperature and the depth of the water in the tank. A colder external temperature will result in a shorter supercooling process and a lower maximum supercooling degree, whereas a deeper external water layer will prolong the main supercooling process and reduce the maximum supercooling value. It has been demonstrated that the residual supercooling temperature Tr ranges from 0 °C to −0.01 °C and that the maximum supercooling degree T

_{min}of the supercooling process spans from −0.03 to −0.1 °C [8,9,24,25,80,81].

## 4. Frazil Ice Generation and Evolution

#### 4.1. Nucleation

#### 4.2. Crystal Growth

#### 4.3. Secondary Nucleation and Flocculation

^{2}/s

^{3}. The analysis reveals that an increase in the turbulence intensity causes an increase in the number of ice crystal collisions, which increases the size of the flocs, but there is also a higher turbulence intensity that overwhelms the strength of the ice particles and causes fragmentation, limiting the growth of the average floc size. This finding is also supported by the photos captured by Clark’s experiments: flocs created in high-turbulence-intensity water are denser in their early and late phases of growth and have a constrained final size. On the other hand, it has been found that condensates generated in water with low levels of turbulence intensity are initially sparser and more fragile, and as they grow larger and become more porous over time.

## 5. Frazil Ice Movement and Distribution

#### 5.1. Uprise Movement and Vertical Distribution

#### 5.2. Frazil Ice Accumulation

_{i}/Q). Hou et al. [125] studied the effect of bridge piers on ice wave migration, derived empirical equations to predict the thickness of ice waves, and investigated the thickness of wave crests and the migration rate of ice waves.

#### 5.3. Attachment to Underwater Objects

## 6. Conclusions

- (1)
- The need for more applications of signal acquisition systems and image acquisition systems in field observations, and a close link between data from signal measurement systems and specific characteristics from image acquisition technology.
- (2)
- The need for a rational approach to determining the evolution of frazil ice generation by the water temperature subcooling process.
- (3)
- The need for numerical models for the initial nucleation process of frazil ice.
- (4)
- The need for new observation techniques to observe the dynamic processes of frazil ice collisions; then, more studies and new models on the secondary nucleation and flocculation of frazil ice.
- (5)
- The need for more studies on frazil ice movement and distribution and new three-dimensional frazil ice movement and distribution models for simulating frazil ice at complex cross sections of rivers and channels.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Schematic (side view) of the turbulent glass water tank designed by McFarlane [20].

**Figure 4.**Schematic (overhead) view of the runway-type circulating water tank designed by Richard [23].

**Figure 5.**Schematic (side view) of the reverse rotation flume device designed by Clark [25].

**Figure 7.**Conceptual diagram of frazil ice as observed in the laboratory [19].

No. | Year | Researcher | Formula | Remark |
---|---|---|---|---|

1 | 1994 | Svensson [77] | $\frac{\partial {V}_{i}}{\partial t}=\frac{{N}_{u}\frac{{k}_{w}}{l}\left({T}_{i}-{T}_{w}\right){A}_{ice}{n}_{i}}{\left({\rho}_{i}L\right)}$ | ${A}_{ice}$ is the active freezing area per crystal, ${A}_{ice}=2\pi {r}_{i}d$ ${\rho}_{i}$ is the density of ice; L is the latent heat of ice. |

2 | 1995 | Hammar [71] | $\frac{\partial {C}_{k}}{\partial t}+{U}_{i}\frac{\partial {C}_{k}}{\partial {x}_{i}}=\frac{\partial}{\partial {x}_{i}}\left(\frac{{\vartheta}_{t}}{{\sigma}_{c}}\frac{\partial {C}_{k}}{\partial {x}_{i}}\right)-{\omega}_{k}\frac{\partial {C}_{k}}{\partial {x}_{3}}+{S}_{{c}_{k}}+{S}_{floc,k}$ ${S}_{{c}_{k}}=\left(\frac{{S}_{k-1}}{\u2206{v}_{k-1}}-\frac{{S}_{k}}{\u2206{v}_{k}}\right){v}_{k}$ $\u2206{v}_{k}={v}_{k+1}-{v}_{k}$ | ${C}_{k}$ is the volumetric concentration of frazil in the k-th size fraction; ${\omega}_{k}$ is the frazil buoyant velocity of the k-th size fraction; ${S}_{{c}_{k}}$ is the source term due to the thermal growth of frazil ice; ${S}_{floc,k}$ is the source/sink term due to secondary nucleation and flocculation. |

No. | Year | Researcher | Formula | Remark |
---|---|---|---|---|

1 | 1994 | Svensson [77] | $\begin{array}{c}\frac{d{n}_{i\text{-}sec}}{dt}\\ \left(1\le i\le N\right)\end{array}=\begin{array}{c}{{\displaystyle \sum}}_{j=2}^{N}{\alpha}_{j}{n}_{j}\\ \left(i=1\right)\end{array}-\begin{array}{c}{\alpha}_{j}{n}_{j}\\ \left(2\le i\le N\right)\end{array}$ ${\alpha}_{j}=\frac{\overline{n}{\mathsf{\Delta}}_{i}}{dt}$ ${\Delta}_{i}={U}_{r}\pi {r}_{i}{}^{2}dt$ ${U}_{r}=\sqrt{{U}_{l}{}^{2}+{U}_{rise}{}^{2}}$ ${U}_{l}={\left(1/15\right)}^{0.5}{\left(\epsilon /\nu \right)}^{0.5}d$ $\overline{n}=\mathrm{min}\left({\displaystyle \sum}_{i=1}^{n}{n}_{i},{\overline{n}}_{max}\right)$ | N represents the number of particle size groups; i represents the i-th size $\mathrm{class};d{n}_{i\text{-}sec}$ is the number change in the i-th particle size group due to secondary nucleation; ${\Delta}_{i}$$\mathrm{is}\mathrm{a}\mathrm{crystal}\mathrm{in}\mathrm{relative}\mathrm{movement}\mathrm{to}\mathrm{the}\mathrm{fluid}\mathrm{that}\mathrm{will}\mathrm{sweep}\mathrm{a}\mathrm{volume};{U}_{r}$ is the relative velocity; ${U}_{rise}$ is the gravitational rise’s velocity; $\epsilon $ is the turbulent dissipation rate; $\nu $ is the kinematic viscosity; d is the crystal diameter; $\overline{n}$ is the average number of crystals per unit volume; |

2 | 1995 | Hammar [71] | $I\left({v}_{i},{v}_{j}\right)=\underset{{v}_{i-1/2}}{\overset{{v}_{i+1/2}}{{\displaystyle \int}}}\underset{{v}_{j-1/2}}{\overset{{v}_{j+1/2}}{{\displaystyle \int}}}Z{C}_{E}\left({v}_{i},{v}_{j}\right)d{v}_{i}d{v}_{j}$ ${C}_{E}\left({v}_{i},{v}_{j}\right)=0.5{\rho}_{i}\frac{{v}_{i}{v}_{j}}{{v}_{i}+{v}_{j}}\left[b{\left({v}_{i}^{1/3}+{v}_{j}^{1/3}\right)}^{5}\frac{{\langle \epsilon \rangle}^{3/2}}{{\vartheta}^{3/2}}{E}_{sh}+0.00076{\left(\frac{g}{\vartheta}\frac{\left|{\rho}_{i}-{\rho}_{w}\right|}{{\rho}_{w}}\left|{v}_{i}^{2/3}-{v}_{j}^{2/3}\right|\right)}^{3}{\left({v}_{i}^{1/3}+{v}_{j}^{1/3}\right)}^{2}{E}_{dr}\right]g\left({v}_{i}\right)g\left({v}_{j}\right)$ | ${v}_{i}$$\mathrm{and}{v}_{j}$$\mathrm{are}\mathrm{the}\mathrm{sizes}\mathrm{of}\mathrm{the}\mathrm{collision}\mathrm{particles};I\left({v}_{i},{v}_{j}\right)$$\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{nuclei}\mathrm{produced}\mathrm{per}\mathrm{unit}\mathrm{time};Z\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{nuclei}\mathrm{produced}\mathrm{per}\mathrm{unit}\mathrm{collision}\mathrm{energy};{C}_{E}$ is the rate of collisional energy transfer to the crystals per unit volume of fluid; $b=0.0066/{K}_{u}{}^{3/4}$$;{K}_{u}$$\mathrm{is}\mathrm{the}\mathrm{kurtosis}\mathrm{of}\mathrm{the}\mathrm{velocity}\mathrm{derivative};g\left({v}_{i}\right)$$\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{density}\mathrm{functions};{E}_{sh}$$\mathrm{and}{E}_{dr}$ are the collision efficiency functions for differential rising and turbulent shear, respectively; |

3 | 1974 | Evans [107] | $\dot{N}=S\left(\u2206T,c,P/{V}_{c},r,\dots \right)\dot{{E}_{t}}$ $\dot{{E}_{tg}}={K}_{g}{f}^{2}\left(0\right){\overline{r}}^{10.42}{{\displaystyle \int}}_{0}^{\infty}{T}_{g}\left({r}_{2}^{\prime}\right)exp\left(-{r}_{2}^{\prime}\right)d{r}_{2}^{\prime}$ ${K}_{g}=0.09{\rho}_{i}{\left[\frac{\left({\rho}_{f}-{\rho}_{i}\right)}{{\rho}_{f}}{\left(\frac{{\rho}_{f}}{\mu}\right)}^{0.6}\right]}^{2.15}$ ${T}_{g}\left({r}_{2}^{\prime}\right)={{\displaystyle \int}}_{0}^{\infty}{t}_{g}\left({r}_{1}^{\prime},{r}_{2}^{\prime}\right)exp\left(-{r}_{1}^{\prime}\right)d{r}_{1}^{\prime}$ ${t}_{g}\left({r}_{1}^{\prime},{r}_{2}^{\prime}\right)={\left(\left|{{r}_{1}^{\prime}}^{1.14}-{{r}_{2}^{\prime}}^{1.14}\right|\right)}^{3}{\left({r}_{1}^{\prime}+{r}_{2}^{\prime}\right)}^{2}\frac{{{r}_{1}^{\prime}}^{3}{{r}_{2}^{\prime}}^{3}}{{{r}_{1}^{\prime}}^{3}+{{r}_{2}^{\prime}}^{3}}$ $\dot{{E}_{tt}}={K}_{t}\left(P/{V}_{c}\right){f}^{2}\left(0\right){\overline{r}}^{8}{{\displaystyle \int}}_{0}^{\infty}{T}_{t}\left({r}_{2}^{\prime}\right)exp\left(-{r}_{2}^{\prime}\right)d{r}_{2}^{\prime}$ ${K}_{t}=3.3{a}^{3/2}\left(\frac{{\rho}_{i}}{{\rho}_{f}}\right)$ ${T}_{t}\left({r}_{2}^{\prime}\right)={{\displaystyle \int}}_{0}^{\infty}{t}_{t}\left({r}_{1}^{\prime},{r}_{2}^{\prime}\right)exp\left(-{r}_{1}^{\prime}\right)d{r}_{1}^{\prime}$ ${t}_{t}\left({r}_{1}^{\prime},{r}_{2}^{\prime}\right)={\left(\left|{{r}_{1}^{\prime}}^{2/3}-{{r}_{2}^{\prime}}^{2/3}\right|\right)}^{3/2}{\left({r}_{1}^{\prime}+{r}_{2}^{\prime}\right)}^{2}\frac{{{r}_{1}^{\prime}}^{3}{{r}_{2}^{\prime}}^{3}}{{{r}_{1}^{\prime}}^{3}+{{r}_{2}^{\prime}}^{3}}$ | $\dot{N}$ is the nucleation rate; $S\left(\u2206T,c,P/{V}_{c},r,\dots \right)$ is the number of crystals generated per unit of collision energy and is expected to be a function of parameters including supercooling $\u2206T$, the salt concentration $c$, the crystal size $r$, and the agitation power $P/{V}_{c}$. The author believes that the mixing power can be replaced by hydraulic conditions such as the turbulence intensity or dissipation rate in natural water flow calculations. $\dot{{E}_{t}}$ is the rate of the energy transfer to the crystals by a collision. $\dot{{E}_{tg}}$ and $\dot{{E}_{tt}}$ are the components of $\dot{{E}_{t}}$ contributed to by collisions between crystals driven by gravity and turbulence, respectively. $f$ is the crystal frequency distribution; $\overline{r}$ is the average crystal size; ${r}_{1}^{\prime},{r}_{2}^{\prime}={r}_{1}/\overline{r},{r}_{2}/\overline{r}$; ${r}_{1}$, ${r}_{2}$ are the crystal radii of crystal–crystal collisions; ${\rho}_{f}$ and ${\rho}_{i}$ are the densities of the fluid and crystal, respectively. |

No. | Year | Researcher | Formula | Remark |
---|---|---|---|---|

1 | 1994 | Svensson [76] | $\begin{array}{c}\frac{d{n}_{i\text{-}flo}}{dt}\\ \left(1\le i\le N\right)\end{array}=\begin{array}{c}-{\beta}_{i}{n}_{i}\\ \left(i=1\right)\end{array}\begin{array}{c}+\delta {\beta}_{i}{n}_{i}\\ \left(2\le i\le N\right)\end{array}$ ${\beta}_{i}={\alpha}_{floc}\frac{{r}_{i}}{{r}_{1}}$ | $d{n}_{i\text{-}flo}$ is the number change in the i-th size $\mathrm{class}\mathrm{due}\mathrm{to}\mathrm{flocculation};\delta $ is the ratio between the volumes of particles of two neighboring radius intervals; ${\alpha}_{floc}$$\mathrm{is}\mathrm{a}\mathrm{calibration}\mathrm{parameter};{r}_{1}$ is the radius of ice particles in the minimum radius group; |

1995 | Hammar [70] | ${F}_{n}=\beta \left({v}_{i},{v}_{j}\right)E\left({v}_{i},{v}_{j}\right){\varphi}_{i}{\varphi}_{j}$ $f=\frac{{v}_{k+1}-{v}_{merge}}{{v}_{k+1}-{v}_{k}}$ $f=\frac{{v}_{merge}}{{v}_{m}}$ ${v}_{merge}={v}_{i}+{v}_{j}-n{v}_{1}$ | ${F}_{n}$$\mathrm{is}\mathrm{the}\mathrm{instantaneous}\mathrm{expected}\mathrm{number}\mathrm{of}\mathrm{collisions}\mathrm{between}\mathrm{all}\mathrm{particles}\mathrm{in}\mathrm{the}i\text{-}\mathrm{th}\mathrm{size}\mathrm{class}\mathrm{and}\mathrm{the}j\text{-}\mathrm{th}\mathrm{size}\mathrm{class}\mathrm{per}\mathrm{unit}\mathrm{volume}\mathrm{per}\mathrm{unit}\mathrm{time};{\varphi}_{i}$$\mathrm{and}{\varphi}_{j}$ are the number concentration of the i-th- and j-th-sized particles, respectively, where each collision per unit volume reduces the local number concentration of i-th- and j-th-class particles by one; ${v}_{merge}$ is the new particle volume due to each collision; ${v}_{1}$ is the volume of the ice nucleus size class. |

No. | Year | Researcher | Technique and Method | Laboratory or Field | Size of Frazil Ice (Diameter) |
---|---|---|---|---|---|

1 | 1950 | Schaefer [28] | Microscope | Laboratory | 1–5 mm |

2 | 1952 | Arakawa et al. [29] | The shadow photograph method | Laboratory | 0.1–3 mm |

3 | 1983 | Osterkamp et al. [51] | Camera | Field | Mostly 0.1 to 1 mm, up to 3–5 mm |

4 | 2006 | Clark et al. [80] | Image acquisition and post-processing technique | Laboratory | Approximately 0.3–5 mm |

5 | 2012 | Ghobrial et al. [37] | Microscope equipped with a digital camera | Laboratory | 0.25–4.25 mm |

6 | 2015 | McFarlane et al. [9] | Image acquisition and post-processing technique | Laboratory | 0.022–5.5 mm |

7 | 2017 | McFarlane et al. [52] | Image acquisition and post-processing technique | Field | Mean of 0.32–1.2 mm |

8 | 2019 | McFarlane et al. [53] | Image acquisition and post-processing technique | Field | 0.034–5.83 mm |

9 | 2022 | Richard et al. [23] | Image acquisition technique | Laboratory | Approximately 1 mm |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Lian, J.; Zhao, X.; Guo, Q.; Yang, D.
Advances in Frazil Ice Evolution Mechanisms and Numerical Modelling in Rivers and Channels in Cold Regions. *Water* **2023**, *15*, 2582.
https://doi.org/10.3390/w15142582

**AMA Style**

Chen Y, Lian J, Zhao X, Guo Q, Yang D.
Advances in Frazil Ice Evolution Mechanisms and Numerical Modelling in Rivers and Channels in Cold Regions. *Water*. 2023; 15(14):2582.
https://doi.org/10.3390/w15142582

**Chicago/Turabian Style**

Chen, Yunfei, Jijian Lian, Xin Zhao, Qizhong Guo, and Deming Yang.
2023. "Advances in Frazil Ice Evolution Mechanisms and Numerical Modelling in Rivers and Channels in Cold Regions" *Water* 15, no. 14: 2582.
https://doi.org/10.3390/w15142582