Collapse of turbulence in atmospheric flows turbulent potential energy budgets in a stably stratified couette flow

Transcription

1 Eindhoven University of Technology BACHELOR Collapse of turbulence in atmospheric flows turbulent potential energy budgets in a stably stratified couette flow de Haas, A.W. Award date: 2015 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 Collapse of Turbulence in Atmospheric Flows Turbulent Potential Energy Budgets In a Stably Stratified Couette Flow Eindhoven, R-1866-S Written by Alex W. de Haas University of Technology Eindhoven Surpervised by Bas J.H. van de Wiel Ivo G.S. van Hooijdonk University of Technology Eindhoven

3 Abstract A phenomenon often encountered in nature, is the fact that, winds may become weak in the evening in case of clear sky conditions. With the weakening of the wind also turbulence may cease. This phenomenon is often referred to as the collapse of turbulence. Recently, this collapse was explained by the so-called maximum sustainable heat flux theory (van de Wiel et al 2012 [7]). Here two other viewpoints to this phenomenon are studied. These will be studied in an idealised model of the atmospheric boundary layer during nighttime. The first theory that is studied is the turbulent kinetic energy (TKE) perspective, which is referred to as the classical theory. Here buoyancy is considered a loss term per definition. In the second perspective, the total turbulent energy (TTE) perspective, buoyancy is not considered a loss term per definition. In this theory buoyancy is considered a transition from kinetic energy to potential energy. In contrast to the maximum sustainable heat flux theory neither of the two studied theories are directly linked to the collapse phenomenon. 2

4 Contents 1 Introduction 4 2 Theory Turbulence near walls The role of stratification Turbulent kinetic energy Richardson numbers Turbulent potential energy Dimensionless energy equations Total turbulent energy Dimensionless height Experimental setup The model Preliminary testing Analysis Results and discussion Preliminary testing: mean profile stratified case Preliminary testing: TKE budget in neutral case Cases with cooling: a TKE perspective Stably stratified couette flow: a TTE perspective Conclusions 26 3

5 1 Introduction When a turbulent flow is cooled it is able to become laminar as a consequence of increasing density gradients. This phenomenon, often referred to as a collapse of turbulence, has often be encountered in nature during nighttime. The study of this phenomenon is relevant for the prediction of fog and extreme frost. Van de Wiel et al (2012) [7] has found a plausible theory, the maximum sustainable heat flux theory, that might be able to explain a collapse of turbulence. In this research, however, two completely different candidate theories as to explain this phenomenon are studied. At first the classical theory will be analyzed. Classically turbulence is described in terms of turbulent kinetic energy (TKE), where buoyancy is considered a loss term. In the classical theory a collapse is anticipated when dissipation of TKE exceeds its production. Recently however Zilitinkevich et al (2007) [8] has proposed a new perspective. In this work it was argued that the TKE energy loss due to buoyancy is not per definition lost. It is a transition to turbulent potential energy (TPE), which is a part of the total turbulent energy (TTE). Therefore they argued that turbulence should be described by the sum of TKE and TPE. The goal of this research is to compare the classical and proposed theory by studying the energy budgets in an idealised flow system for different cooling rates. 4

6 2 Theory 2.1 Turbulence near walls Turbulence is characterized by internal swirling motion of fluids. Those swirling fluid elements are referred to as turbulent eddies. As a result of these eddies fluid velocity and pressure vary as a function of place and time. At the large scale large eddies are produced as a result of instabilities of the mean flow. These eddies live in the macrostructure, where frictional effects are negligible. These large eddies become unstable and break up into multiple smaller eddies. The smaller eddies break up in smaller eddies as well, and so on until the microstructure is reached. This process is known as the cascade-process. In the microstructure viscous friction is dominant. This implies that small eddies dissipate energy due to viscous friction and molecular energy transport processes. In this research turbulence the interest lies in atmospheric flows during nighttime. An idealized model of these flows is analyzed. This idealized model is a channel flow. The turbulence in channel flows is described as wall turbulence in Nieuwstadt (1992) [6]. The flow can be divided in three regions. The largest regime is the core region, which is located far from the wall. In this regime viscous friction is negligible and eddies are scaled with the geometry of the flow. When we get closer to the wall the size of eddies are limited. Here the physics of the core region doesn t apply anymore. This region is called the wall region. Even more closer to the wall the flow is not able to be turbulent anymore. This region is called the viscous sublayer. The viscous friction is not negligible here, but dominant. This implies that turbulent energy will dissipate here. Note that the flow is not laminar in the viscous sublayer, because this would imply that no fluctuations exist here. However in this region velocity fluctuations do exist as a result of turbulence above the viscous sublayer. 2.2 The role of stratification As mentioned before in this research an idealized model of the atmospheric flow during nighttime is analyzed. This idealized model is a stably stratified couette flow. In a stratified flow the dynamics are directly influenced by gravity. This implies that density gradients influence the flows characteristics. Density gradients are a result of temperature gradients, because cold air has a larger density than warm air. Therefore it costs energy to lift cold air above warmer air. For the same reason it costs energy to push warm air downward. This implies that when temperature gradients exist it costs more energy to mix up the air. Temperature gradients are therefore able to suppress turbulence. In particular temperature gradients are able to decrease turbulent length scales. During nighttime temperature gradients are created through cooling of the ground. It costs more energy to mix up this stratified air. Thus buoyancy has an oppressing effect on turbulence during nighttime. Note that during daytime the air is warmed up by the ground and buoyancy stimulates turbulence. From observations we know that buoyancy is able to shut off turbulence completely during nighttime. This is called a collapse. In case of collapsed turbulence heat transport by turbulence is suppressed. As a result air near the surface becomes colder. This can result in condensation of water vapor, creating a fog layer. Recent studies (van de Wiel et al (2012) [7]; 5

7 Figure 1: Radiation fog in Topanga Canyon. van Hooijdonk et al (2015) [3]) have found a plausibel theory, that could explain a collapse of turbulence. This theory will be explained briefly in this section. When a flow is in thermal equilibrium, a collapse does not occur. In order to be in thermal equilibrium the maximum of the turbulent heat flux of the flow should be greater than the energy demand from the ground. In other words the turbulence should be able to transport enough heat downward to compensate the heat loss near the ground. Figure 2: Sketches of magnitude of turbulent heat flux and radiative heat loss for cases with strong winds (left) and weak winds (right). Stronger winds produce more turbulent energy and are therefore able to provide more downward turbulent heat transport. Thermal balance with the ground can be reached if the downward heat flux is enough to compensate the energy demand of the ground. Therefore balance is more likely to be reached, when the wind velocity is large. This theory is illustrated in figure 2. In case 1 there is a strong wind; the flow is able to provide enough energy to overcome the radiative heat loss. This flow is thus able to reach a thermal equilibrium. In case 2 the situation for a weak wind is sketched. Here the flow is not able to provide enough energy to overcome the radiative heat loss. Thus here a collapse will certainly take place. Note that finally the air will become very cold near the surface. As a result the radiative heat loss will diminish and soil heat flux will increase. The heat flux is limited in the absence of vertical temperature gradients. But when 6

8 Figure 3: Sketch of the turbulent heat flux curve vs stability for weak winds (red), the minimum ( critical ) wind (black), and strong winds (blue). The energy demand is given by the dashed line. Here, Q N is the net radiative loss at the surface. The soil heat flux G is incorporated in the demand. For strong stability levels, Monin-Obukhov similarity is no longer valid(e.g., Mahrt (2014) ); as a result, the shape of the curves becomes uncertain. (Figure was originally presented in van Hooijdonk et al (2015) [3]). vertical temperature gradients become too large vertical mixing is oppressed. This results in lower downward heat flux as well. Thus the downward heat flux maximizes at intermediate stability. This theory is illustrated in figure 3. When the maximum of the turbulent heat flux is smaller than the energy demand of the ground, the temperature gradient will increase, as well as the stability level. In that case all turbulence will be suppressed eventually, which is called a collapse. The described maximum heat flux theory is a plausible explanation for a collapse. In this research an alternative explanation for the collapse phenomena will be discussed. This theory will be based on the energy budgets of turbulence, which are subject of the next section. 2.3 Turbulent kinetic energy In this section we will characterize turbulence by its kinetic energy. The description of turbulence in terms of its kinetic energy budgets is considered the classical theory of turbulence energy in this report. It will be motivated that turbulent energy dissipates by internal friction and buoyancy. The buoyancy flux is the second factor, that may dissipate TKE. This represents the work done by the flow against the gravity field. In this research we consider atmospheric flows with local wind velocity U = (u, v, w) in the cartesian coordinate system (x, y, z), pressure p and potential temperature θ. Throughout this report the Einstein convention will be used, where u 1 = u, u 2 = v, u 3 = w, x 1 = x, x 2 = y and x 3 = z. Our search for a relation for the turbulent kinetic energy starts at the Boussinesq approximation. The Boussinesq-relations are approximated solutions to the Navier-Stokes equations. In this approximation the flow is considered incompressible (see Nieuwstadt (1992) [6]). In Nieuwstadt (1992) [6] these equations are given as u i x i = 0, (1) 7

9 u i t + u u i j = θ g i 1 p + ν 2 u i, (2) x j T 0 ρ 0 x i x 2 j θ t + u θ j = κ 2 θ, (3) x j x 2 j In which ρ 0 is the reference density, ν is the viscosity, κ is the diffusion coefficient, T 0 is the reference temperature and g i is the gravitational constant in the i-direction. The gravity is per definition directed in the negative z-direction, thus g i = gδ i3. All quantities can be split in a mean and a fluctuation, for example u i = u i + u i, where u i is the mean velocity and u i is the fluctuation. This notation for the velocities is substituted in equation (2). The result is averaged in space, which gives the equation for the average flow. Consequently the equation for the average flow is subtracted from the original equation (2) to receive the equation for the fluctuation in the velocity field: u i t + u u i j + u u i j + u i u j x j x j x j u i u j x j = 1 ρ 0 p i x i + g T 0 θ δ i3 + ν 2 u i x 2 i (4) From this result we can deduce a relation for the turbulent kinetic energy E k = 1 2 u 2 i. By multiplying equation 4 by u i and consequently averaging the result we arrive at DE k Dt = u u i i u j + g w x j T θ + ( u j 0 x E k 1 p j ρ u j + ν E ( k u ) ν i 0 x j x j ) 2 (5) In which E k = 1 2 u 2 i and D/Dt / t + u j / x j. From left to right the terms behind the equal sign are identified as the production of turbulent kinetic energy, the buoyancy term, the three transport terms and the dissipation of kinetic turbulent energy. The three transport terms are the transport of energy due to velocity fluctuations, pressure fluctuations and viscosity respectively. All in all we can conclude that there is one mechanism produces TKE: the production by the mean flow. There are two loss terms for TKE: the dissipation by internal friction and the work done by the flow against the gravity field. The remaining three terms are the transport terms, which solely distribute energy, but do not create or dissipate energy. 2.4 Richardson numbers Under stably stratified conditions (Nieuwstadt (1992) [6]) the transport terms in equation (5) are usually small. When these are neglected, the remaining terms are the production term and the two loss terms. The TKE perspective suggests that buoyancy is able to cause a collapse. Therefore it seems relevant to calculate the magnitude of buoyancy relative to the production of kinetic energy. From (5) it follows that the fraction of produced energy that is lost due to buoyancy is equal to R f = g T 0 w θ u i u j u i x j (6) where R f is the flux Richardson number. A flow is considered neutral when the buoyancy is zero (or negligible). Thus a flow is neutral when R f = 0. When the buoyancy term is 8

10 positive,it produces TKE and add to the shear production term. Therefore a flow is considered stable when R f > 0. When R f > 1, buoyancy consumes more energy than the flow produces. The flow will then certainly become laminar. However in practice the energy dissipation consumes energy as well. Therefore a collapse will generally occur for smaller Richardson numbers. In this research we re interested in the cooling cases, where the buoyancy is negative. The TKE perspective suggests that the flux Richardson number can be used to predict whether a flow is going to collapse or not. In practise the flux Richardson number is of less value, because it is difficult to measure directly. As an approximation an alternative dimensionless number may be used (de Bruin 1998 [1]): R i = g T 0 θ z ( ) u 2 (7) z where R i is the gradient Richardson number. Atmospheric observations suggest that a flow becomes unstable when the gradient Richardson number exceeds the threshold value of 0.2 (de Bruin 1998 [1]). Thus a collapse will be expected when R i 0.2. In that case all turbulence will cease. Note that this is threshold value is not hard proven, but an estimation based on observations. Within the approximation of the K-theory one can show that P r t = R i R f (8) where P r t is the turbulent Prandtl number, which is the fraction of momentum eddy diffusivity and heat transfer eddy diffusivity: P r t = k M k H (9) where k M and k H are the eddy diffusivities for momentum transfer and heat transfer respectively. The flux and gradient Richardson numbers are functions of the distance to the wall. To determine these functions, the quantities need to be known for many points in a certain interval. In practical experiments this is difficult to achieve. Therefore an discretized version of the gradient Richardson number is introduced in de Bruin 2006 [1]: R b = (z 2 z 1 ) g T 0 θ ( u) 2 (10) where R b is the bulk Richardson number. Note that in the limit z = z 2 z 1 0 the Bulk Richardson number is equal to the gradient Richardson number. Because of this relation with the gradient Richardson number, the same threshold value can be used. Thus we expect that all turbulence is stopped when R b O(0.2). The limits in the Richardson numbers determining whether a collapse occurs is a conjecture supported by experimental observations. These limits are not hard proven, but still the Richardson numbers are able to provide some insight in the stability of flows. This theory is based on the classical view, that buoyancy is a loss term. This means that energy needed to overcome buoyancy is lost per definition. In this theory turbulent potential energy is not taken into account and might therefore be inaccurate or incomplete, because part of the buoyancy loss might come back in the system. 9

11 2.5 Turbulent potential energy In Zilitinkevich et al 2007 [8] it is argued that the TKE perspective in its sole form is incomplete. He states that the loss in turbulent kinetic energy by buoyancy is equal to the gain in turbulent potential energy. It is argued that this potential energy is also part of the total turbulent energy. This doesn t mean that the buoyancy term can simply be scrapped from the kinetic energy equation. Instead turbulence has to be described by the summation of TPE and TKE budgets. In this section the energy budgets of the TPE will be deduced. In order to generate an equation for the turbulent potential energy we first define E θ = 1 2 θ 2. Note that this quantity is not really an energy quantity. This is however a convenient starting point to deduce an equation for the turbulent potential energy, as will become clear later. The equation for the potential temperature energy is derived using the Boussinesq approximation as was done for the kinetic energy of turbulence. The splitted notation for quantities, θ = θ + θ, is substituted in equation (3). The result is averaged, which gives the equation for the average flow. The equation for the average flow is subtracted from the original equation (3) to receive the equation for the fluctuation in the potential temperature field: θ t + u θ j + u θ j + u j θ u j θ = κ 2 θ x j x j x j x j From this result we can deduce a relation for the potential temperature energy E θ. By multiplying equation 11 by θ and consequently averaging the result we arrive at DE θ = w Dt θ θ z 1 u j θ θ + κ 1 2 θ θ ( ) θ 2 2 x j 2 x 2 κ (12) j x j Note that E θ doesn t contain a dimension of energy. The first term behind the equal sign is identified as the buoyancy term. From this we can derive the equation of the turbulent potential energy E P (containing the same unit as E k ). Because the buoyancy terms in both equation must be equal, but with opposite sign, it follows that: E P = g ( ) 1 θ E θ (13) T 0 z By dimension analysis it is confirmed that E P has energy dimensions (not shown). The equation for the turbulent potential energy therefore becomes: ( DE P = g ( ) 1 w Dt T θ g θ 1 u j θ θ g ( ) 1 ) θ κ 1 2 θ θ 0 T 0 z 2 x j T 0 z 2 x 2 g ( ) 1 ( ) θ θ 2 κ j T 0 z x j (14) The first term behind de equal sign is the buoyancy term, the second term is the transport of potential energy due to velocity fluctuations, the third term is the transport of potential energy due to molecular conduction and the last term is the dissipation of turbulent potential energy. Zilitinkivic et al (2007) [8] obtained the same result for the turbulent potential energy, except for one term. The transport due to viscosity is neglected in the paper by Zilitinkivic et al (2007) [8]. Later in this report it is discussed whether it s good approximation to neglect this term or not. 10 x 2 j (11)

12 2.6 Dimensionless energy equations The budgets derived in sections 2.3 and 2.5 are in this research analyzed in numerical simulations of couette flows are analyzed. In order to adequately calculate and compare these budgets, the equations have to be made dimensionless. To achieve this we will first have to define the characteristic velocity, length and temperature scale, which are U 0 and h and T respectively (see figure 4). The characteristic temperature scale is the turbulent heat flux divided by U 0 : T = H 0 ρc p U 0 where c p is the heat capacity of air at constant pressure. Figure 4: Sketch of flow profile in a couette flow (both take the same form). h and U 0 are the characteristic length and temperature scale respectively, where 2h is the height of the canal and U 0 is the velocity of the moving plate. Using the described characteristic scales three dimensionless parameters describing the flow can be deduced: Re = V 0h ν = 2500 which is the Reynolds number, P r = ν κ = 1 which is the molecular Prandtl number and h = gh 0T T 0 U 2 0 which is the cooling rate (different for each case). By replacing all quantities by dimensionless quantities times a characteristic scale we arrive at the dimensionless equation for the turbulent kinetic and potential energy: DE k Dt = u u i i u j + h w x θ + ( u j j x E k 1 p j ρ u j + 1 E k ) 1 ( u i 0 Re x j Re x j ) 2 (15) 11

13 DE P Dt ( ) 1 θ 1 = h w θ h z 2 u j θ θ +h x j ( ) 1 θ 1 z 2 1 θ θ Re P r x 2 x 2 j ( ) 1 ( ) θ h 1 θ 2 z Re P r x j Note that all quantities here are dimensionless. The same symbols are used in both the dimensionless and dimensional energy equations. In the remainder of the report all energy budget equations will be dimensionless, while using the same symbols for simplicity reasons. These energy equations are still general. In the studied simulations all pressure gradients are zero and therefore the transport due to pressure fluctuations is small (negligible) in all cases. Additionally the entire xy-plane is assumed to be statistically uniform. In other words the average of all velocity and temperature fluctuations are independent of x and y. Therefore all gradients with respect to x and y of averages are equal to zero per definition. Furthermore the average flow is directed in the x-direction. Therefore the gradients of the average flow in the y- and z-direction are zero. Under these conditions eq. (15) reduces to where DE k Dt (16) = P k + βf z + T k + D k ɛ k (17) P k = w u u z is the production term, βf z = h l w θ is the buoyancy term, T k + D k = 1 2 w u 2 i z Re 2 u 2 i z 2 are the transport of energy due to velocity fluctuations and viscosity respectively and ɛ k = Re( 1 ) u 2 i x j is the dissipation of kinetic energy. Using these approximations and the dimensionless parameters of the flow the Richardson numbers for this flow become: R f = h w θ w θ = h (18) u i w u i P x k j R i = h θ z ( ) u 2 (19) z R b = (z 2 z 1 )h θ ( u) 2 (20) Similarly the equation for the turbulent potential energy, eq. (16), reduces to DE P Dt = βf z + T P + D P ɛ P (21) 12

14 where ( ) 1 θ 1 T P + D P = h w θ 2 ( ) 1 θ 1 + h z 2 z z 2 are the transport terms for potential energy and ( ) 1 ( ) θ 1 θ ɛ P = h 2 z Re x j is the dissipation of potential energy. 2.7 Total turbulent energy 1 Re 2 θ 2 z 2 Now that the energy budgets equations are derived, the turbulent energy can be described by the sum of its potential and kinetic energy. In this report this theory will be referred to as the proposed theory. The sum of these budgets will be called the total turbulent energy (TTE) and is defined as (Zilitinkevich et al (2007) [8]) ( E T = E K + E P = 1 ( ) 1 ) u 2 i + h θ θ 2 z 2 (22) To find an equation for the total turbulent energy, we simply add equation (21) to (17): DE T Dt = DE k Dt + DE P Dt = P k ɛ k ɛ P + (T k + D k + T P + D P ) (23) Note that the buoyancy term cancels here. The terms between brackets on the right hand side of the equation are the transport terms. In stably stratified flows these are generally very small and over the entire regime of the canal they don t contribute to energy production or loss. If we neglect these terms for the moment the only terms that remain are the production term and the two dissipation terms. This implies that there s only one mechanism, that is responsible for the production of turbulent energy. The shear production remains the same as in the TKE approach. From a (classical) TKE perspective energy is lost directly through kinetic energy dissipation and buoyancy. Now in the new proposed theory, the buoyancy is considered not a loss term. However in that case we must include another loss term, which is the dissipation of potential energy. 2.8 Dimensionless height In the previous sections the energy budget equations for the classical and proposed theory are derived. To gain confidence in the simulations results will first be compared with the literature. The calculated kinetic energy budgets for the case without buoyancy, will be compared with Nieuwstadt 1992 [6]. As Nieuwstadt (1992) [6] presents the turbulent energy budgets in wall units, the same normalization is adopted. In Nieuwstadt 1992 [6] the height is normalized using the viscosity and the friction velocity u : z + = z u ν (24) 13

15 where z + is the dimensionless height. The friction velocity is obtained from the turbulent shear stress τ: τs u = ρ u w v w (25) in which τ s is the wall shear stress and ρ is the density of the flow. Note that the viscosity terms are neglected in the wall shear stress here. By substituting eq. (25) in eq. (24) we arrive at the following dimensionless result: z + = zre u w v w (26) 14

16 3 Experimental setup The goal of this research is to compare the (classical) TKE picture and the new perspective of TTE. The classical theory says buoyancy is a loss term per definition. In the proposed theory buoyancy is a transition to TPE, which can also contribute to turbulence. To compare the theories the energy budgets are analyzed in an idealized model flow for the evening transition in the atmospheric boundary layer. In the atmosphere the flow field around sunset may be considered neutrally stratified. After sunset it becomes stably stratified due to net emission of radiation from the surface. The emission of radiation is compensated by energy provided by the flow. The air close to the ground loses energy to the ground (which in turn cools by emission of longwave radiation) and will become cooler. Thus in a cooled flow temperature gradients exist. Because cooler air is heavier than warm air, more energy is needed to mix up the air in the cooling case. Classically it s assumed that all the extra energy needed to mix up the air is lost; buoyancy is a loss term. In Zilitinkevich et al (2007) [8] it s suggested that the kinetic energy loss is equal to the potential energy gain. They suggest that since no energy is lost by the mixing of air, where temperature gradients apply, a collapse can not occur. However in previous studied models, collapses of turbulence is actually observed. Therefore in this study all kinetic and potential energy budgets will be calculated for the idealized model. The goal is to provide deeper insight in the proposed theory in Zilitinkevich et al (2007) [8] and in the classical theory. It is attempted to either disprove the proposed theory or match the proposed theory with the classical theory. 3.1 The model As mentioned before an idealized flow will be used in this research. This configuration was proposed by Ivo G.S. van Hooijdonk. It is an idealization of the atmospheric boundary layer. All buildings and objects in the atmosphere are neglected. The atmosphere is modelled as an empty box, where shear stress does occur on the ground, but is not a function of place (the box is uniform). The flow is forced by a infinite plate that moves with constant speed at a certain height above the ground. Onlt the wind relative to the ground is of interest. Therefore the model is symmetric and both the top and bottom of the box move at constant speed in opposite direction. Thus the simulations models two infinite plates differing a height z, that move with constant velocity U 0 in opposite direction. The described model is referred to as a couette flow. A sketch of the wind velocity and the shear stress in this flow is illustrated in figure 5. Now since we cannot model an infinite sized box, the box doesn t have infinite length. However in the model boundary effects are not desirable. Therefore the boundary conditions are periodic. The model calculates per time step certain quantities per blok, where it uses the previous values of surrounding blocks. To overcome the fact that the blocks at the sides of the box aren t completely surrounded by neighbours, the blocks from the other side are used there as neighbours. In other words the model doesn t stop at the edges, but continues at the other side. The plates lie in the xy-plane per definition. The distance between them is 2h in the z-direction. The width of the studied regime in the x- and y-direction is 10h. The regime is divided in 360, 360 and 180 grid points in the x- y- and z-direction respectively. For these 15

17 Figure 5: Sketch of flow profile in a couette flow (both take the same form). grid points the local velocities and potential temperature are calculated at each time step. Also the pressure is calculated for each grid point. The described model is simulated for different cooling rates h, which are in the order of All cases are run for 300 dimensionless time units. The described quantities are calculated and stored every 4 dimensionless time units. The potential and kinetic energy budgets described in section 2.6 are calculated using these outputs. Since the model is statistically homogeneous in the x- and y- direction, the energy budgets are averaged over these directions. The result is a matrix with the energy budget values as a function of time and height z. Note that the model is symmetric in the middle of the canal (at z = 1h). Therefore these budgets are only calculated for the bottom half of the canal. 3.2 Preliminary testing The programmed model should be verified in some sense to be sure that the theory is implemented as intended. To this end a first-order comparison with Monin-Obukhov similarity theory is made. The latter is a semi-empirical theory founded on vast amount of atmospheric observation and data. A reasonable agreement between the model results and and Monin- Obukhov predictions will indicate that the non-dimensionalisation is performed correctly and that correct order of magnitude is obtained in the results. This provides confidence in the model for further analysis. First the mean velocity profiles are compared to Monin-Obukhov similarity theory. The mean flow profiles are compared to the log-linear and van Driest wind profiles. Consequently the energy budgets are compared qualitatively with earlier results by others. The kinetic energy budgets for a simulation without cooling (no buoyancy) are compared with Nieuwstadt 1992 [6]. 3.3 Analysis Once the model is verified the classical and proposed theory will be analyzed. First the cooling cases, where no collapse occurs, are evaluated. They will be treated classically. Thus the turbulence is described by its kinetic energy only and the buoyancy is considered a loss term. Also the Richardson numbers will be analyzed for these cases. Consequently we will add the potential energy budgets to test the proposed theory of total turbulent energy. By analyzing the potential energy budgets we will try to find out what happens with the potential energy that is created by buoyancy. Hereby we will try to 16

18 either match the classical TKE theory with the new TTE theory or disprove the new TTE theory. After analyzing the non-collapse cases, we will briefly study a collapse case. Again the kinetic and potential energy budgets will be determined, as well as the Richardson numbers. All theories potential for predicting a collapse will be discussed. Finally the potential of the TKE and TTE perspective as a predictive tool to anticipate collapse at strong cooling will be discussed. 17

19 4 Results and discussion 4.1 Preliminary testing: mean profile stratified case The first goal of this research is to gain confidence in the performed simulations. Therefore the flow profile as a function of the height in the canal is plotted (see figure 6). In the same figure the log linear profile is plotted. Also the log linear profile with van Dries-extension is plotted. This extension corrects for the effects of viscosity, which are dominant close to the wall (Van Driest (1956) [2]). These are the typical flow profiles for this type of canal flow. The figure shows that the simulated mean profile is described remarkably well by those simple laws. This implies that the non-dimensionalisations are done correctly in the model and that realistic results are obtained. This gives confidence in the correctness of our simulations. Note however that this doesn t necessarily mean that all turbulent properties (higher-order moments) are realistic as well. Figure 6: The flow profile determined for the simulations for the neutral case is graphed here, together with the van Driest profile and Log-linear profile. (Figure provided by I.G.S. van Hooijdonk through personal communication.) 4.2 Preliminary testing: TKE budget in neutral case Next a second test is performed in order to see if also the budgets of TKE are realistic. The case without cooling is ideal to test this. Here all temperature gradients are equal to zero. This implies that the buoyancy term and all turbulent potential energy budgets are equal to zero. The goal is to analyze the budgets during the steady state. When the steady state is reached the results at different time steps are statistically identical. We will analyze the time-average over a certain time interval when a steady state is reached. By analyzing the mean velocity speed and the root mean square of the velocities, we were able to deduce the time needed for a steady state to be achieved. For the case without cooling, but also for the other non-collapse cases, it appears that a steady state is reached after t=

20 Additional analysis (not shown) revealed that extending the time interval has no noticeable effect on the results. This tells us that the interval is long enough and it will therefore be used throughout this report for the analysis of steady states. The time-averaged kinetic energy budgets (see section 2.3) in the steady state without cooling are determined and plotted against the dimensionless height (see section 2.8). Consequently they are compared qualitatively with the results from literature presented in Nieuwstadt (1992) [6]. The result is depicted in figure 7. Figure 7: Left: TKE energy budgets as a function of the dimensionless height z + for the neutral case (no cooling), where magenta is P k, blue is T k, green is D k and red is ɛ. All Energy budgets are averaged over the last 16 time steps. The values in the grey area are less accurate due to boundary effects. Right: TKE energy budgets as a function of the dimensionless height z + as presented in Nieuwstadt (1992) [6]. These budgets are also for a neutral case (no cooling). The form of the graphs seems very similar. Thus our results qualitatively agree with the found solutions in Nieuwstadt (1992) [6]. Only qualitative comparison can be made here due to the use of a different set-up and Reynolds number. For the smallest values of z + (grey area) however the model seems less accurate. This might be a result of boundary effects. Furthermore because the space is divided in blocks, this causes inaccuracy. The flow changes the most the closest to the wall, so there the results are less accurate. Also numerical differentiation causes inaccuracy, which is more dominant close to the wall as well due to larger gradients. It gives confidence that similar results are found as in the literature. In the results we see that the maximum of the production of TKE is near the wall, but not right at the wall. Turbulence is produced by the shearing mean flow, which is the largest at the wall. However very close to the wall large eddies can not exist. Consequently the maximum of the kinetic energy production is close to the wall, but not right at the wall. As for the dissipation of kinetic energy the maximum is expected at the wall. As discussed in the theory close to the wall only small eddies can live. For small eddies viscous friction is dominant. Thus where many small eddies exist, a lot of energy is dissipated. Over the full domain the transport terms only redistributes TKE. The total calculated integral under over the determined transport terms is negative (but small). In Nieuwstadt (1992) [6] we see that D k continues to rise close to the wall. In the obtained results D k goes to zero here. This might cause the difference and is probably a result of the boundary effects 19

21 or the inaccuracy in numerical differentiating. All in all we can conclude that the results in the core region agree with the literature and our qualitative expectations. However the results very close to the wall might be less accurate. Therefore in the rest of this report the kinetic energy budgets are assumed to be realistic at least for the core and logarithmic region, but less accurate in the vicinity of the wall 4.3 Cases with cooling: a TKE perspective Next the cooled couette flow is studied with respect to its TKE budgets (i.e. the classical view).this implies that the buoyancy term, which is now not equal to zero, will be considered a loss term in the TKE budget. For various cooling rates TKE budgets are studied. The focus lies on cases below the critical cooling rate for ceasing turbulence. In these cases the energy supply is enough to overcome the energy demand at the ground. Therefore these flows will reach a steady state. We studied the time evolution of the mean velocity speed and root mean square of the velocities to verify that a steady state has been reached. As for the neutral case a steady state is reached at t=240. From the neutral case we know that averaging over the last 60 time units, is enough for a statistical convergence. The kinetic energy budgets as described in section 2.3 are plotted for cooling rate of h = (weak cooling rate) and h = (moderate cooling rate) in figure 8. In the remainder of this report these cases will be called hl3 and hl7 respectively. Hl7 case is the closest to a collapse that will be analyzed in this report. Figure 8: TKE energy budgets as a function of the dimensionless height z + for cooling rates h = (left) and h = (right), where magenta is P k, black is the buoyancy, blue is T k, green is D k and red is ɛ k. All Energy budgets are averaged over the last 16 time steps. The values in the grey area are less accurate due to boundary effects. The first thing that stands out is that the budgets in figure 8 appear rather similar to the neutral case. The main difference is in magnitude. For other sub-critical cooling rates similar results are obtained. Again the difference is mainly in scale, all terms except the buoyancy term decrease over the entire region for increasing cooling rate. Only the buoyancy term increases for higher cooling rates. For larger cooling, temperature gradients increase, therefore it is to be expected that the buoyancy destruction increases. Due to stratification the integral length scale of turbulence decreases. Transport of energy is suppressed and production decreases (though shear increases somewhat). As seen in figure 8 the buoyancy is very small in comparison to the production and the 20

22 dissipation. The buoyancy reaches a maximum in the middle of the canal (z = 1h). This maximum is per definition equal to h. Even at the middle of the canal the buoyancy is relatively very small. In figure 9 a part of figure 8 (hl7 case) is enlarged. From this graph it becomes clear the buoyancy does play a role in the core region of the canal, although it s still relatively small, about 10%. By increasing the cooling rate further to a collapse occurs (not shown). As even for the critical case the contribution of buoyancy to the total budget is still less than 10% (van Hooijdonk personal communication), it is unlikely that the energy loss due to buoyancy is the direct cause for a collapse. Obviously the cooling is the reason for the collapse, but the reason might be more indirect, as for example explained by the maximum sustainable heat flux theory of van de Wiel et al (2012) [7]. Figure 9: Production (magenta) and absolute value of buoyancy (black) term as a function of the dimensionless height z + for the hl7 case around the middle of the canal. In figure 9 the absolute value of the buoyancy and production are illustrated as a function of the dimensionless height. The buoyancy is taken positively here, because negative values are not allowed on a log-scale. From figure 9 it becomes clear that the buoyancy consumes about 10% of the production of kinetic energy in the hl7 case. This would imply that the flux Richardson number is about 0.1. This case is close to a collapse, but still the flux Richardson number is half the threshold value of 0.2 (de Bruin (1998) [1]). Let s take a closer look at the Richardson numbers for this case and the hl3 case. The flux Richardson number and gradient Richardson number are graphed in figure 10. As seen in the graph both Richardson numbers are below 0.2. Later we will elaborate on this aspect. In figure 11 the values of the gradient and flux Richardson numbers in steady state at the middle of the canal are graphed. We can see that they follow a straight line as a function of the cooling rate. If this trend would continue the gradient Richardson number will approximately equal R i 0.13 at h = , which is the first case where a collapse occurs. In section 2.4 it was discussed that the flux Richardson number and the gradient Richardson number are more or less equivalent when P r t 1. As the graphs for R f and R i appear very similar, suggest that this is true for this case. The turbulent Prandtl numbers for the same cooling rates as above are plotted in figure 12. Appart from the region in vicinity of the wall the Prandtl number is about 1 in both cases. 21

23 Figure 10: Flux Richardson number (blue) and gradient Richardson number (red) as function of the dimensionless height z + for cooling rates h = (left) and h = (right). The values in the grey area are less accurate due to boundary effects. Figure 11: Flux (blue) and gradient (red) Richardson number end values as a function of the cooling rate h. The last values at the middle of the canal are used. This holds for other analyzed non-collapse cases as well. The grey area is less accurate partly because turbulence is suppressed due to viscous effects. But the main reason is probably that both the flux and gradient Richardson number become very small in the vicinity of the wall. So that P r t is rather poorly defined from (8). In the hl3 case a dip is identified in the turbulent Prandtl number in the middle of the canal. This is the result of the discussed dip in the gradient Richardson number, which we could not clarify. However for this couette flow we may generally assume that it is indeed reasonable to treat the flux and gradient Richardson number as similar. In section 2.4 a third Richardson number was discussed, namely the bulk Richardson number. The bulk Richardson numbers are calculated for the lower half of the domain. For the hl3 and hl7 case the bulk Richardson numbers over this region equal about 0.01 and 0.03 respectively. Those values are rather low in view of the range for R i and R f (figure 10). Partly this could be explained because it represents an average over a large region. In figure 10 we see that the flux and gradient Richardson numbers are very small close to the wall. This will probably hold for the bulk Richardson number as well, resulting in smaller values. This raises the question whether the bulk Richardson number is useful to analyze in this type of flow. The bulk Richardson number is probably of more use in regions with uniform shear, because averaging over non-uniform regions provides less useful information. 22

24 Figure 12: Turbulent Prandtl number as function of the dimensionless height z + for cooling rates h = (left) and h = (right). The values in the grey area are less accurate due to boundary effects. We will briefly study Richardson numbers in a collapse case to gain more insight in the predictive potential of the Richardson numbers. In figure 13 the flux and gradient Richardson number are illustrated as a function of time for the hl9 case at approximately a quarter height of the canal. At different heights similar results are found. The results have low physical value, because turbulence has collapsed in this case and the Richardson numbers are a measure for turbulence. Both have unpredictable forms and sporadic peaks. The reason for these peaks is not determined in this research and might be subject for future research. Figure 13: Flux Richardson number (left) and gradient Richardson number (right) as a function of the time for the hl9 case at height z = 5/9h. The gradient Richardson number never exceeds the threshold value of 0.2 (according to de Bruin (1992) [1]). The flux Richardson number does, but only sporadic. Therefore it s not certain whether the threshold value of 0.2 is entirely accurate. Since this is not the main focus of this research, this will not be studied in more detail. This might be an interesting subject for future research however. 4.4 Stably stratified couette flow: a TTE perspective Until now in this report the cooling cases are analyzed from a TKE perspective and the buoyancy was considered a loss term. In this section an effort is made to analyze the total turbulent energy proposal (see section 2.7). By adding the TKE and TPE budgets the buoyancy term cancelled. In return we obtained two potential energy transport terms and a potential energy dissipation term. Thus the difference between the classical theory and the proposed theory is the buoyancy loss versus the potential transport plus potential energy dissipation. 23

25 The starting point is the same as for the classical case, where only TKE budget was analyzed: the cooling cases without collapse. As discussed in section 4.3 the analyzed cases are sure to have reached a stable state at time step 240. By averaging over the last 60 time steps (240 until 300) sufficient statistical convergence is obtained.. The TKE and TPE budgets are plotted for the hl3 and hl7 cases in figure 14. Plotting the TKE and TPE budgets in one figure is not useful, because the TPE budgets are substantially smaller than the TKE budgets. Figure 14: Top: TKE energy budgets as a function of the dimensionless height z + for cooling rates h = (left) and h = (right), where magenta is P k, black is the buoyancy, blue is T k, green is D k and red is ɛ k. All Energy budgets are averaged over the last 16 time steps. The values in the grey area are less accurate due to boundary effects. Bottom: Potential energy budgets as a function of the dimensionless height z + for cooling rates h = (left) and h = (right), where black is the buoyancy, blue is T P, green is D P and red is ɛ P. The values in the grey area are less accurate due to boundary effects. The buoyancy term is the production term for the turbulent potential energy. As seen in the graph for increasing cooling, the buoyancy increases as well. This is to be expected and was already discussed in section 4.3. The transport terms are relatively small. In Zilitinkevich et al 2007 [8] T P is ignored. In these results it s shown however that this term is not negligible. Instead it s even more significant than the other transport term D P. In both presented cases it becomes clear that the dissipation of turbulent potential energy is of the same order as the production of turbulent potential energy. This also holds for the other non-collapse cases (not shown). Therefore the assumption that energy lost due to buoyancy is not returned to the flow, is correct. The old perspective however, which states that the energy is lost immediately, is not correct. The energy is passed on from TKE to TPE, where it is dissipated via thermal diffusion. All in all the buoyancy energy is still lost (indirectly). Therefore the TTE concept is not in disagreement with the fact that turbulence may collapse for strong cooling. Yet, neither with TKE or TPE framework quantitative 24

26 predictions of such collapse can be made in agreement with couette results. At present the maximum sustainable heat flux theory seems to be the best candidate for such purpose. 25

27 5 Conclusions In this research an idealized model of atmospheric flows was analyzed. The calculated energy budgets from this model were compared to results found in the literature. Qualitatively these agreed very good with each other. Only close to the wall our simulations seem less accurate. This is probably the result of boundary conditions and inaccuracy in numerical differentiation. While studying the TKE budgets for the cooling cases we found that budgets keep the same form as in the neutral case. But the amplitude of all budgets decrease for increasing cooling rate. The buoyancy term is the only one that gains in amplitude. But even when we approach a collapse very closely the buoyancy is relatively small in comparison to the production of kinetic energy. Thus the loss of TKE due to buoyancy is not able to directly explain the collapse of turbulence. For the hl7 case, which is close to a collapse, the Richardson numbers are approximately 0.1. Even for the hl9 case, where a collapse occurs, the Richardson numbers do not exceed the threshold value of 0.2. Therefore the correctness of this threshold value can be questioned. Furthermore we saw that the Richardson numbers take unpredictable forms for the hl9 case, which is probably a result of the fact that turbulence has ceased in this case. To gain more inside in the proposed theory the TPE budgets of the cooling cases were analyzed. This showed that all buoyancy is dissipated as turbulent potential energy. This implies that the classical approach, that buoyancy is a loss term, seems correct. However analyzing TPE still has its utility, because it gives more inside in the mechanism through which buoyancy energy is lost. All in all we may conclude that the classical theory seems correct, but incomplete. Buoyancy term is indeed a loss term, but it is lost through dissipation of TPE. But still a collapse cannot be explained by the energy budgets. This can potentially be done however by the maximum heat flux principle described in section