PLANT BIOPHYSICS - Y. I. Posudin - 2004

II. TRANSPORT PROCESSES IN THE SOIL-PLANT-ATMOSPHERE SYSTEM

10. MOMENTUM TRANSPORT

10.1. BOUNDARY LAYER

Class="center">10.1.1. CHARACTERISTICS OF THE Laminar Boundary Layer

Energy absorbed by a plant is utilized in heating, photochemical reactions, and evaporation. The heating of a plant or its environment occurs via Heat transfer through thermal conduction and convection, whereas evaporation is associated with Water vapor transport via Transpiration or carbon dioxide via Photosynthesis. Heat and mass transport are driven by molecular diffusion through a thin layer of air near the leaf surface, known as the boundary layer. The characteristics of this boundary layer depend on the viscous properties of air and momentum transport by internal friction (viscosity) forces.

Let us consider the air boundary layer along a smooth surface formed during airflow along this surface. If the flow regime is laminar (moving air layers do not mix), momentum transfer occurs between individual molecules. Indeed, as an air stream moves over a solid surface, its velocity increases with distance from the surface, creating a velocity gradient (Fig. 10.1). This gradient, which can be considered linear in the first approximation, arises from frictional forces against the surface. Frictional forces also occur between air layers moving parallel to one another with different scalar velocities. The faster-moving layer exerts an accelerating force on the slower-moving layer, and conversely, the slower-moving layers decelerate the faster-moving ones. The resulting frictional forces act parallel to The surface of the layers. Due to the velocity gradient, momentum mV is transported. The rate of momentum transport τ is determined by the expression:

where η = ρv is the dynamic viscosity coefficient, ρ is the average air density, and v is the kinematic viscosity coefficient. The momentum flux is considered positive if it is directed toward the solid surface (see Fig. 10.1).

Fig. 10.1. Scheme of momentum transport by a moving air stream to a stationary surface.

The thickness of the boundary layer has certain dimensions determined by the transition of the laminar flow regime into a turbulent one, characterized by layer mixing. The flow regime is characterized by the Reynolds number, defined by the formula:

where η is the dynamic viscosity of air, ρ is the air density, and D is the characteristic dimension of the system (in the case of a leaf, the distance from the edge). The transition from laminar to turbulent flow is determined by the critical Reynolds number Re_cr: if Re < Re_cr, the flow is laminar; at Re > Re_cr, the flow is turbulent.

Calculations of the characteristics of the laminar boundary layer, particularly its thickness, On the surface of bodies of various shapes have been performed by integrating the boundary layer equations [Prandtl, 1920]. The approximate calculation method proposed by T. von Kármán [Karman, 1954] can be formulated as follows: "Let us isolate a region in the flow bounded by a fixed surface of arbitrary shape. Flow particles passing through this region change their momentum. It is known that momentum equals the product of mass and velocity. The increase in momentum of all particles passing through a given region per unit time can be expressed as the difference between the momentum of particles leaving the region per unit time and the momentum of particles entering the region during the same time. This change in momentum per unit time equals the inertial forces and must be in equilibrium with the external forces acting on the surface of the region or within it." Detailed calculations can be found in the work of Eckert and Drake (1961). The main results of applying these principles to a two-dimensional flow along a flat surface are presented below.

Fig. 10.2. Airflow velocity profiles above a flat surface: a - uniform airflow; b - laminar boundary layer; c - turbulent boundary layer.

When considering a uniform airflow (Fig. 10.2, a), a laminar boundary layer is formed as it passes over a flat surface. The velocity distribution curve in this layer is distorted (Fig. 10.2, b). Within the boundary layer, the flow velocity increases from 0 to a steady, stationary value. Further on, a turbulent layer arises (Fig. 10.2, c), in which velocity increases more rapidly. The magnitude of the boundary layer thickness δ increases proportionally to the square ROOT of the distance l from the edge of the surface [Eckert and Drake, 1961]:

where v is the kinematic viscosity (v = η/ρ), l is the distance from the edge of the surface, and V_st is the stationary velocity.

It is practical to express the latter formula in a dimensionless form.

Currently, METABOLISM/2.html">THE CONCEPT OF the equivalent boundary layer thickness δ' is used, which is determined using the method shown in Fig. 10.3. Rectangle abed is equal to the area of the hatched figure bounded by the velocity distribution curve, the ordinate, and the asymptote.

Fig. 10.3. Equivalent boundary layer thickness δ*_gr (explanation in text).

10.1.2. Frictional Forces

In the case of a laminar boundary layer on a surface swept by an air stream, a tangential frictional stress arises, caused by frictional forces. This stress is equal to:

where

If the length of the surface along the flow is equal to L, then the average value of the frictional stress over the entire surface is:

where

Using electrical analogies, one can imagine that momentum transfer is counteracted by a certain resistance. If the path length from the air layer having a velocity V to the surface where V = 0 is defined as l, then the frictional stress will be equal to:

where R is the momentum transfer resistance, which is determined by the following equation:

10.1.3. Form Drag

If a certain body is placed in an air stream, momentum transfer occurs not only due to frictional forces directed parallel to the layer surfaces and the resulting velocity gradient, but also because a force arises in the direction of the flow, which is called form drag. This force is caused by different pressure values on the front and rear sides of the body as the flow moves around it, and depends on the body's shape and orientation. If the initial momentum of the body is pV and the average velocity change is V/2, then the rate at which the flow decreases its momentum will be equal to and the drag force is defined as c_m0.5pV², where c_m is the drag coefficient. In practice, both form drag and surface friction are taken into account simultaneously by introducing the total drag coefficient where S is the surface area washed by the flow.

To study momentum transfer processes on the leaf surface, leaf models—so-called replicas made of aluminum [Thom, 1968]—were used. The dimensions of the replica are shown in Fig. 10.4, along with the dependence of the total drag coefficient (taking into account that the leaf has two surfaces). It can be seen that the drag is minimal when the replica is positioned parallel to the air stream. Applying electrical analogues, the momentum transfer process along the flow is described by the resistance Since form drag is proportional to V² and surface friction drag is proportional to V^0.5, the total drag coefficient can be written as where n is a constant.

Fig. 10.4. Shape and dimensions of the leaf model.

10.1.4. Resistance to Plant Particles

The drag force acting on a particle (assuming it is spherical) can be analyzed by considering certain possible scenarios related to the ratio between the particle radius and the mean free path l of the gas molecules surrounding the particle:

1. The particle radius r is significantly smaller than the mean free path l of the gas molecules. In this case, compared to the gas molecules, the particle acts as a giant molecule acted upon by small gas molecules. Since the particle is moving, the difference between the resulting forces acting on the front and rear surfaces of the particle represents the drag force. This force is determined by the expression:

where n is the number of gas molecules, m_g is the mass of the gas molecules, is the average velocity of the gas molecules, and V is the velocity of the particle.

Thus, in this situation, the drag force is proportional to the particle velocity and its surface area.

2. The particle radius r exceeds the mean free path λ of the gas molecules, but the Reynolds number Re characterizing the relative motion of the particle and the gas molecules is small, i.e., In this case, internal friction forces arise between the particle and the gas; for a spherical particle of radius r, the viscous drag force is determined by Stokes' law:

where η is the coefficient of internal friction (viscosity).

3. The particle radius r exceeds the mean free path λ of the gas molecules, but Re > 1. In this situation, form drag dominates, and the drag force is defined as follows:

where S is the cross-sectional area of the particle and ρ is the gas density.

The dependence of the total drag coefficient c_оп for a particle of diameter l on the Reynolds number Re_v = Vd/v is shown in Fig. 10.5. For small values of Re_ч, the drag coefficient c_оп, obtained from equations (10.10) and (10.11), is linearly dependent on (Re_ч)^-1:

Fig. 10.5. Dependence of the total drag coefficient c_оп of a particle of diameter d on the Reynolds number Re = Vd/V.

A more precise expression that corresponds to the real situation was found empirically:

For high values of Re_ч > 10³, which correspond to the turbulent regime, c_оп remains constant, and the drag force is proportional to pV² · S.

Example. Determine the drag force for a cylindrical spore of the phytopathogenic fungus Helminthosporium maydis, given a cylinder diameter d = 20 µm, density ρ = 1.2 kg·m^-3, total drag coefficient c_оп = 4, Reynolds number Re_ч = 10, and wind speed v = 10 m·s^-1.

Solution. Using equation (10.11):

10.1.5. Vertical Wind Speed Profile Near the Earth's Surface

Wind speed in all directions of a three-dimensional coordinate system consists of the sum of the mean velocity and random velocity fluctuations V_і^і:

where i = x, y, z, with

Under natural conditions, a turbulent boundary layer forms near the Earth's surface. Let us consider a homogeneous horizontal surface large enough for a well-developed boundary layer to form. Assume that the air flow propagates in only two directions: horizontal and vertical.

The theory for determining the mean wind speed profile in a turbulent boundary layer was developed by Prandtl [Prandtl, 1920]. Let us examine the Main principles of this theory. Prandtl drew an analogy between turbulent and molecular motions, equating the mean free path in the kinetic theory of gases with the mixing length l. Thus, it can be assumed that fluctuations in the horizontal velocity V at level z correspond to the arrival of a particle from a lower (z-l) level. Consequently, the fluctuations of velocity in the horizontal direction are determined by the equation:

These horizontal velocity fluctuations are related to the vertical velocity fluctuations U'. The vertical momentum flux is determined by the equation:

where ρ is the air density.

For A large number of air particles, the average vertical momentum flux will be equal to:

If the height above the Earth's surface is significant, we can assume that whence:

Here is measured in (kg·m·s^-1)·m^-2·s^-1.

Substituting V’ from equation (10.15) into the final expression, we obtain:

Assuming that velocity fluctuations are identical in all directions,

we can derive the expression:

Prandtl also hypothesized that fluctuations are proportional to the distance from the Earth's surface:

where k = 0.4 m·s^-1 is the von Kármán constant.

This indicates that the momentum flux density is constant. Consequently, the frictional stress caused by the interaction of wind with the Earth's surface is given by:

where V* is the friction velocity, a parameter with the dimension of velocity, defined by the formula:

Combining equations (10.21)-(10.24), we obtain the differential equation describing the vertical wind speed profile:

which, upon integration, yields the expression for the logarithmic wind profile in the turbulent boundary layer:

where V* is the friction velocity, which is constant within the 50-100 m atmospheric surface layer; z₀ is the roughness length; z is the height; and 0.4 m·s^-1 is the von Kármán constant.

The values of V and z_g can be easily determined experimentally from a semi-logarithmic plot of wind speed versus height: the intersection of the linear fit with the coordinate axes gives the numerical values of these parameters (Fig. 10.6). The roughness length values for various natural surfaces are listed in Table 10.1.

Table 10.1. Roughness length values for various natural surfaces

Fig. 10.6. Vertical wind speed profile near the earth's surface in a semi-logarithmic coordinate system.

If the earth's surface is covered with vegetation of height h, it can be assumed that the zero plane level is displaced upward by a height d < h. Then the equation for the vertical profile takes the following form:

where d is the zero-plane displacement.

Equation (10.26) holds true provided that z ≥ z₀-d. The parameter V* is proportional to the wind speed at height z; furthermore, it depends on the friction of the air flow against the surface. The roughness parameter z₀ has the dimension of length, but it should not be interpreted as a directly measurable physical quantity. This parameter is determined empirically by measuring wind speeds at various heights and extrapolating the linear relationship into the region where V(z) = 0. The intersection of this line with the ordinate axis gives the value of lnZ₀. For uniform agricultural fields, the roughness parameter z₀ and the zero-plane displacement d can be estimated from the vegetation height h. For a field of moderate plant density, the following relations can be used:

Example. What is the friction velocity if the wind speed at a height of 2 m over a potato field of height 60 cm is 4 m·s^-1, and the zero-plane displacement is 1.2 m?

Solution. From Table 10.1, we find the roughness parameter value z₀ = 0.04 m. Substituting the numerical data into equation (10.27), we obtain:

Hence:

10.1.6. Aerodynamic Resistance

The momentum flux caused by the horizontal momentum gradient per unit volume can be rewritten in terms of electrical analogies:

Where R_i is the aerodynamic resistance, which is defined by the expression:

where V(z) is the wind speed at height z.