in collaboration with
The University of Texas at Dallas Department of Mechanical Engineering
MSc in Aeronautical Engineering
AERODYNAMICS OVER COMPLEX
TERRAINS AND ITS EFFECTS ON
WIND FARM POWER PRODUCTION:
AN LES STUDY
Advisors:
Prof. Maurizio Quadrio Prof. Stefano Leonardi
Candidate: Matteo Zangrandi
University of Texas at Dallas". A special thank to Prof. Leonardi, for giving me this opportunity and make me feel part of the team since the first day I have arrived there.
I want to thank all the guys of the Waterview Science and Technology Center, in particular Federico and Umberto for teaching me all what I needed and for their fundamental help throughout my stay at the laboratory.
Un grazie speciale a Matilde, per essermi stata accanto, nonostante gli 8000 chilometri di distanza e le 6 ore di fuso orario. Grazie per avermi insegnato a non mollare, per avermi supportato, ma soprattutto sopportato sempre, per darmi ogni giorno, anche a distanza, la serenità di cui ho bisogno. Grazie ai miei Genitori, per il continuo e costante supporto, per la fiducia incondizionata, per aver tentato in ogni momento di stemperare la tensione in questi 5 anni e per avermi dato la possibilità di dare il mio meglio in ogni occasione. Il supporto economico è stato indispensabile, ma è solo una minima parte di ciò che avete fatto per me.
Grazie a mia sorella Anna per essermi sempre stata d’esempio: grazie al tuo impegno e alla tua determinazione ho capito non ciò che dovevo, ma ciò che volevo diventare.
Grazie a mia Nonna per aver avuto sempre più certezze di me riguardo agli esiti degli esami.
Grazie agli Amici, quelli che posso non vedere per molto tempo e ritrovare come non fosse passato un giorno.
Grazie a tutte le persone che ho incontrato e avuto modo di conoscere durante questi 5 anni: ai colleghi e amici di Skyward, ai professori del Po-litecnico e ai compagni di corso, in particolare a Luca, Giorgia, Zighi, Zizzo (e Nini), Riccardo, Johnny, Giuseppe e Ale. Ognuno di loro mi ha lasciato
Professor Quadrio hanno pubblicato negli anni molti articoli su roughness e wall turbulence: questo comune interesse è il filo conduttore del lavoro. In particolare, gli obiettivi della tesi sono approfondire la conoscienza del flusso su una parete con roughness tridimensionale e studiare l’interazione di un flusso proveniente dalla roughness con una wind farm. La decisione di esten-dere l’analisi con le simulazioni sulla wind farm è motivata dall’esperienza pluriennale del Professor Leonardi nel campo della wind energy. Queste simu-lazioni sono inoltre effettuate con lo stesso codice utilizzato per le simusimu-lazioni sulla roughness.
Il lavoro può essere suddiviso in due parti: la prima si concentra su simu-lazioni di una parete con roughness tridimensionale sinusoidale. La geometria della parete è descritta dalla seguente funzione:
y = a sin 2π λx x sin 2π λz z + 1 (1) dove λx e λz sono le lunghezze d’onda della roughness in direzione
rispetti-vamente longitudinale e trasversale al flusso. Sono stati analizzati sei diversi casi con tre differenti λx e due λz. Inoltre, gli stessi casi sono stati testati
a differente Reb: a basso Reb con una coarse Direct Numerical Simulation
sono stati calcolati i profili di velocità mediati nel tempo e nello spazio, la roughness function, il coefficiente di resistenza e gli sforzi turbolenti. Per le simulazioni ad alto Reb, sono state impiegate Large Eddy Simulations con
il modello di Smagorinsky per modellare le subgrid scales. Per queste, sono riportati profili di velocità e correlazioni a 2 punti della velocità in direzione longitudinale al flusso. Nella seconda parte, le simulazioni LES sulla wind farm sono state effettuate con l’introduzione del modello a disco attuatore rotante per simulare il rotore con le forze di volume generate in funzione del flusso incidente sul rotore stesso. Nacelle e torre sono modellate come corpi attraverso il metodo dei contorni immersi. Le simulazioni ad alto Reynolds sulla roughness sono state utilizzate come inlet per le wind farm in modo da verificare l’influenza del flusso sul potenza prodotta dalle pale eoliche. Inoltre sono state verificate le correlazioni a 2 punti della velocità longitudinale in varie posizioni all’interno della wind farm.
is mainly focused on simulations on a sinusoidal tridimensional roughness choosen to model the complex terrain varying the streamwise and spanwise wavelength of the geometry. Six different cases have been analyzed at low and high Reynolds number in order to characterize the flow. Simulations at high Reynolds number are then been used as precursor for LES simulations on a wind farm. Results show mean velocity profiles, stresses and 2D corre-lations
The second part analyzes the interaction of the flow coming from the precur-sor with a simple wind farm. Results are analyzed in terms of power recovery of the last rows of the wind farm and correlations of streamwise velocity.
1 Introduction 1
2 Background 5
2.1 Flows over rough walls . . . 5
2.2 Wind turbine model . . . 7
2.3 Governing equations . . . 13
2.4 Numerical Method . . . 14
2.4.1 Discretization of Navier-Stokes equation . . . 14
Linear terms . . . 15
Non-Linear terms . . . 15
Solution and pressure correction . . . 16
Stability and accuracy of the code . . . 20
2.4.2 Immersed Boundary Method . . . 21
2.4.3 Direct Numerical Simulation approach . . . 21
2.4.4 Large-Eddy Simulation approach . . . 23
2.4.5 Filtered Navier-Stokes Equations . . . 24
3 Simulations over complex terrain 29 3.1 Simulation setup . . . 30
3.2 Coarse DNS Simulations at Low Re number . . . 32
3.2.1 Results . . . 32
Mean velocity profile . . . 34
Drag Coefficient . . . 35
Roughness function . . . 37
Velocity fluctuations . . . 41
3.2.2 Large Eddy Simulations at High Re . . . 48
3.2.3 Results . . . 49
Mean velocity profiles . . . 49
Drag Coefficient . . . 50
Two-Point 2D correlations . . . 51 v
4.1.1 Transition from the precursor . . . 57
4.2 Results . . . 58
4.2.1 Power production . . . 59
Wake recovery . . . 60
Two-point correlations of streamwise velocity fluctuations 62 4.2.2 Correlations between velocity at inlet and power. . . . 63
5 Concluding remarks 69
1.1 Rough wall waviness . . . 2
2.1 Nikuradse’s Diagram [54] . . . 6
2.2 Moody’s Diagram [47] . . . 7
2.3 Roughness Function. Adapted by Jiménez [28]. . . 8
2.4 Blade element and actuator forces. . . 9
2.5 Typical wind turbine power curve . . . 10
2.6 NREL 5-MW performance curves. . . 12
2.7 Grid cell with staggered velocities. . . 15
2.8 Geometry with immersed boundary method. . . 22
3.1 Grid spacing in y-direction. . . 31
3.2 Mean velocity profiles . . . 35
3.3 Drag coefficient as function of λx . . . 37
3.4 Diagnostic function with different virtual origins . . . 39
3.5 Roughness function ∆U+ as function of λ x . . . 41
3.6 Logarithmic wall law before and after shift. . . 41
3.7 Logarithmic wall law with pressure-gradient correction . . . . 42
3.8 R.m.s. profiles of streamwise velocity fluctuation . . . 43
3.9 R.m.s. profiles of streamwise velocity fluctuation - 2 . . . 44
3.10 Rms of streamwise velocity fluctuations on Plane XY . . . 46
3.11 Rms of streamwise velocity fluctuations on Plane YZ . . . 47
3.12 Mean velocity profiles . . . 50
3.13 Mean Velocity Profiles of case 1.1 . . . 51
3.14 Drag coefficient Cd . . . 52
3.15 Length and inclination of structures . . . 53
3.16 Contours of correlations on XY plane . . . 54
4.1 Wind farm layout . . . 56
4.2 Grid spacing in y-direction. . . 57
4.3 Cross section comparison . . . 58
4.4 Mean velocity profiles in spanwise direction . . . 59 vii
4.7 Fixed points for correlations in the wind farm . . . 62
4.8 WF 1.1: Contours of correlations on plane XY . . . 65
4.9 WF 3.2: Contours of correlations on plane XY . . . 66
4.10 Points for correlation velocity-power . . . 67
3.1 Setup parameters . . . 30 3.2 Low-Re simulation: Reynolds number and resolution in wall units. 33 3.3 Wall law results . . . 40 3.4 Setup parameters. . . 49 3.5 Drag coefficient values and variations with the Reynolds number. 50 3.6 Two-point 2D correlations results . . . 53 4.1 Parameters of the wind turbine NREL-5MW . . . 56 4.2 Wake recovery data . . . 61
DNS Direct Numerical Simulation LES Large Eddy Simulation r.m.s. Root Mean Square
RADM Rotating Actuator Disk Model ALM Actuator Line Method
ADM Actuator Disk Model TKE Turbulent Kinetic Energy IBM Immersed Boundary Method CFR Constant Flow Rate
CPG Constant Pressure Gradient CPI Constant Power Input Cd Drag coefficient
Introduction
Flow over rough wall has being one of the main topics in turbulence dur-ing years. Some of previous studies have focused on very simple and regular roughness, such as transversal bars or cubes. For example, Leonardi et al. [40] and Furuya, Miyata, and Fujita [18] point out the dependence of drag on the streamwise spacing of transversal square bars and circular rods respectively, while Leonardi and Castro [39] correlated the drag over staggered cubes with their plan density. Mori, Quadrio, and Fukagata [48] investigated the drag reduction effectiveness of uniform blowing over a two-dimensional, but irreg-ular roughness. Other studies [2, 7, 13] have been carried out considering a tridimensional -but still parametrized- roughness. Among those, Bhagana-gar, Kim, and Coleman [2] quantified the effect of "egg-carton" roughness on the flow varying the streamwise and spanwise spacing. The common interest in pursuing a deeper knowledge on tridimensional roughness suggested the collaboration with Prof. Stefano Leonardi and his laboratory at "The Uni-versity of Texas at Dallas" (UTD). We studied a tridimensional sinusoidal roughness varying the streamwise and spanwise wavelength to assess what is the behaviour of the flow and how it is affected by these parameters. The same code used for roughness simulations has been validated during years for wind farm simulations, thus the idea of connecting the two topics. Indeed, this work aims both to characterize the flow over a complex terrain and study the mutual interaction of a wind farm and the turbulent structures generated by the terrains.
As showed in [6], the irregular rough wall can be simplified by applying a low-pass filter to the geometry. Based on this idea, our roughness has been choosen to model in a simple way a generic complex terrain, considering only one Fourier mode for each direction as representative of the geometry.
Figure 1.1: Example of the waviness of the rough wall.
Further knowledge of the flow characteristics over complex terrains may help in understanding the effects of obstacles on the atmospheric boundary layer. This effect is crucial in certain situations. For example, interaction between wind turbines and the surrounding environment is one of the most challenging problems to be studied. It is not always easy to assess the be-haviour of the airflow over hills, buildings and any other obstacle in the neighborhood, but these features may cause a dramatic loss in wind power production. For this reason, most of onshore wind farms around the world are located in plain areas. However, some of those can be found in a hilly environment, such as the facility in Perdigao (Portugal).
The present work can be divided in two parts: in the first part, we carried out several simulations on a complex terrain, modeled as a three dimensional sinusoidal roughness varying wavelengths with fixed height. The simulations ran enough time to allow the flow to be fully developed and to make statistics more reliable. We analyzed and characterized the flow in terms of velocity profiles, rms of fluctuations of streamwise velocity and two-point correlations. Then, we used these results as a precursor for simulations on the wind farm. In the second part, we performed wind farm simulations, choosing the previously obtained results as input. Hence, we were able to link the effects of the turbulent structures generated by a certain terrain on the wind farm and to study the evolution of the coherent structures within the flow.
The following chapter will present the numerical method implemented in the code, as far as the theoretical background necessary to apply it to
two-point correlations and spectra at low Reynolds in order to validate the code, comparing the results with the previous studies. Same simulations are carried out with high Reynolds number and analyzed as well. They are used as precursor for wind farm simulations. Chapter 4 presents the results of the simulations on wind farm mostly about wake recovery, 2D correlations and correlations between power and velocity at the inlet. Conclusions are presented in chapter 5 along with possible future works.
Background
In the following sections, we provide brief introduction on the flows over rough walls. It is also explained how the wind turbines are modeled on the physical side and how the code works from a numerical point of view to solve the Navier-Stokes equations.
2.1
Flows over rough walls
Roughness is one of the most discussed topics in research on turbulent flows. The first historical studies were mainly about pipe flows. Nikuradse [54] conducted his experiments in 1933 on hydraulic effects of uniform sand grain roughness in cylindrical pipes. His data (Figure 2.1) are still considered as the standard in fluid dynamics, even if they were obtained as the result of gluing real sand grains to the pipe walls.
This graph shows the friction factor λ (or f) as function of the Reynolds number and it is parametrized with the relative roughness ǫ
d, where ǫ is
a sort of mean diameter of sand grains and d is the diameter of the pipe. Moreover, there were many attempts to define an empirical law to describe the friction factor, among those Colebrook’s formula (1939) is probably the most well-known. Then, Rouse (1942) and Moody (1944) tried to use the formula to compute the friction factor of several circular commercial pipes and they summarized it in a graph showing the relationship between the same parameters employed by Nikuradse: this gave birth to the famous Moody Chart (Figure 2.2). During the following years, many studies were carried out to extend these results to open-channels.
Research on roughness enlightened also its importance on the wall tur-5
Figure 2.1: Nikuradse’s Diagram [54]
bulence. Some studies suggested that three different regimes may be found depending on k+, (roughness height non-dimensionalize with viscous length
δν):
• k+< 4: smooth wall;
• 4 < k+< 70: transition;
• k+> 70: fully rough wall;
On the contrary, others emphasize the fact that different kinds of roughness exist and the shape of the roughness element has a fundamental role, thus the k+is not sufficient in defining the flow regime. Historically, two kinds of
roughness are described: it can be k-type or d-type, depending on the size of eddies with respect to the size of the roughness element itself.
In the present work, we will describe a complex terrain as a transitionally and fully rough wall. However, one can define an extended wall law (2.1) taking into account roughness.
U+= 1 κlog(y
+
) + A − ∆U(k+) (2.1) where U+is the flow velocity non-dimensionalized by the friction velocity u
τ,
κ is the Von Karman constant, y+ and k+ are respectively the wall normal coordinate and the roughness height both non-dimensionalized by viscous length δν, A is a constant equal to 5.2, ∆U+ is the roughness function. In
Figure 2.2: Moody’s Diagram [47]
Figure 2.3, the ∆U+ is plotted as function of k+. Thus B is the intercept of
the logaritmic law in case of rough wall:
B = A − ∆U(k+) (2.2)
Basically, the extended wall law is still characterized by the logarithmic re-gion, as it was for the smooth wall, but its effects is a shift in the wall law, which can be quantified by the so-called roughness function ∆U+.
Unfortu-nately, this law cannot take into account for different shapes of roughness. In fact, the ∆U+ is strongly dependent on the peculiar case we are considering,
for this reason we are also interested in computing its dependence on the parameters of our complex terrain.
2.2
Wind turbine model
There are several models to account for the effect of the blades on the flow. Among these, we will briefly compare the actuator line method (ALM) and the rotating actuator disk model (RADM), which is implemented by Ciri et al. [10] in this code. In contrast with the actuator disk model (ADM), both the methods preserve the rotation of the blades.
With these models, the actual blades are replaced with the forces they apply to the flow. These forces are applied as a body force fturb
i in the
Figure 2.3: Roughness Function. Adapted by Jiménez [28].
The blade are discretized into a finite number of sections, where the local forces are computed with the local relative velocity, angle of attack and airfoil look-up tables. Hence (Figure 2.4a), the flow velocity relative to the airfoil, Urel, and the angle of attack, α are computed at each section:
Urel= p U2 x+ (Uθ− ωr)2 (2.3) α = arctan Ux Uθ− ωr − φ (2.4)
with φ the local twist angle of the blade, ω the angular velocity of the blades, r the local radius, also the distance of the considered airfoil from the center of the rotor, Ux the streamwise velocity, and Uθ the azimuthal velocity on
the rotor plane (see Figure 2.4(a)).
Discretizing the blades, it is possible to express the lift and drag per unit length of the airfoil as:
L = 1 2ρU 2 relCL(α)c (2.5) D = 1 2ρU 2 relCD(α)c (2.6)
where c is the airfoil chord, CL and CD are the lift and drag coefficients,
which are assumed to be functions of only α. The rotating actuator disk model consists of projecting the aerodynamic force FFF , which is sum of LLL and
(a) Urel Uθ− ωr Ux rotor plane α L D φ (b) 0 20 40 60 80 0 2 4 6 8
Figure 2.4: (a) Blade airfoil: forces and velocity definition; (b) Thrust force
distribution fturb
x in the rotor plane for (left) actuator line model (ALM) and (right)
rotating disk actuator model (RADM). The angular position of the blades and the rotor disk are also shown with dashed white lines. [9]
DDD, onto the flow field, by using a Gaussian function to spread the force over a disk. The Gaussian kernel is given by:
η = exp " − x − xc ǫ 2# · exp " − θ − θblade π/B 2# (2.7) The force fffturb can be expressed as follows:
fffturb= RRFFF η
Aη dA
(2.8) In equation (2.7), x − xc is the relative distance in the streamwise direction
of a generic point from the rotor position (xc), while θ − θbladeis the relative
angular distance from the angular position of the blade θblade. B is the
number of blades; in the present case, B = 3, and the force on each blade is distributed in a sector of 2π/3.
The ǫ parameter for the RADM represents the thickness of the disk. In [44], for the similar actuator line model (ALM), it is suggested to be such that ǫ ≥ 2∆x, where ∆x is the grid cell size in the direction of the rotor axis. This condition should be respected for numerical solution without oscillations. In this case, ǫ = 2∆x1 is used.
The RADM is similar to the ALM [68]. An example of the thrust force distribution on a rotor for the ALM and RADM is shown in Figure 2.4b. The RADM is expected to provide a lower accuracy than the ALM in the repre-sentation of the smallest flow scales (for example, the tip vortices). However, the force spreading in the RADM is smoother than the one of ALM, thus larger time steps can be used in the simulations with the RADM. This leads
to much lower computational costs than with the ALM, given that the CPU time per time step is the same for the two models. As a consequence, a simulation with RADM is about six times faster than using ALM and this is the reason of its choice.
In the present code, the turbines tower and nacelle are represented in the computational grid through the Immersed Boundary method [57], which enables the inclusion of solid bodies inside the computational domain using cartesian grids and avoiding the computationally expensive body-fitted grids. This method has been validated in previous studies both for flow over rough walls [4, 40] than with experiments on a model of wind turbine in a wind tunnel [65]. The results show that inclusion of tower and nacelle in the simu-lation significantly improves the agreement with the measurements. Further validation can be found in [9, 66].
The turbines are driven by their own variable-speed variable-pitch control system based on a simplified two-region architecture [3, 5, 29, 37, 59]. The instantaneous rotational speed of the turbine ω is determined through the angular momentum balance between the aerodynamic torque Taero and the
generator torque Mgen:
I ˙ω = Taero− Mgen (2.9)
where I is the rotor inertia and the ‘dot’ indicates a time derivative. The aerodynamic torque Taerois obtained from RADM, while Mgen is determined
by the control system.
Figure 2.5 shows a typical wind turbine power curve, with the character-istic velocities which separate the different operating regions.
0 0.2 0.4 0.6 0.8 1 Wind speed Vin Vrated Vout P /P ra ted
Figure 2.5: Typical wind turbine power curve
Region I is the region for wind velocity lower than Vin: in such a situation,
the turbine is shut down since there is not enough energy in the incoming wind speed. Between the cut in wind speed, Vin (typically Vin= 3 ∼ 4 m/s),
and the rated speed (typically Vrated= 11 ∼ 14 m/s) the turbine is operating
rated
by the turbine reaches the rated power (Prated). For wind speed larger than
Vrated, the turbine is operating in Region III and the power is limited to the
rated value. This is done to avoid excessive loads on the turbine structure, even if the turbine is working unefficiently. For wind speed larger than the cut out velocity Vout, the wind is too strong and the turbine is shut down to
avoid structural damage. The cut-out velocity is usually around 25 m/s for most modern wind turbines.
Region II is usually the most important region for power optimization study, while in Region III power is saturated by the rated capacity of the machine and cannot be increased (the main concerns in Region III are the loads). In Region II, the control law is proportional to the square of the turbine angular speed ω:
Mgen = kgen· ωgen2 (2.10)
where ωgen = N · ω is the shaft angular velocity, N the gear ratio (N = 97,
for the NREL 5-MW turbine [30]). In equilibrium ( ˙ω = 0), the operating rotational speed is determined by the value of the torque gain kgen.
Blade Element Momentum (BEM) theory [21] can be employed to calcu-late the power coefficient CP:
CP =
Power
1 2ρU∞3A
, (2.11)
where A = πD2/4, assuming an isolated turbine in uniform flow. For
in-stance, Figure 2.6 shows the power and thrust (CT = Thrust/12ρU∞2 A) for
the NREL 5-MW reference turbine [30]. From this figure, CP has its
maxi-mum at the optimal tip speed ratio (TSR) λopt = 7.5. To maximise efficiency,
the torque gain is choosen as: kgen = 1 N2 1 2ρπR 5 CP(λopt) λ3 opt (2.12a) = 2.2 Nm/rpm2 (2.12b)
where λopt = 7.5 for the NREL 5-MW reference turbine, Fig. 2.6.
We consider the torque gain in equation (2.12) equal to the ideal design value. This choice is justified, since for an isolated turbine it is typically kept constant and to the ideal design value, in order to maximize the power obtained in the Region II regime. Changes of torque gain affect the angular
0.0 0.2 0.4 0.6 0.8 1.0 3 4 5 6 7 8 9 10 T SR CP ,C T
Figure 2.6: NREL 5-MW performance curves: thrust (CT, ) and power
(CP, ) coefficients versus TSR. The solid circle (• ) indicates the maximum
efficiency CP,maxobtained at the optimal tip-speed ratio TSR = 7.5
speed of the turbine, because it changes the torque of generator and thus the equilibrium of equation 2.9), as a consequence it changes the operating tip-speed ratio of the turbine.
In Region III, the turbine is operated to produced rated power at constant angular speed (rated angular speed). The torque applied by the generator is determined as follow:
Mgen =
Prated
ωrated
(2.13) where ωrated is the rated speed of the generator (divided by the gear ratio
N ). The controller tunes the blade pitch angle, so that ω = ωratedin Region
III. This implemented with a proportional-integral (PI) controller, which determines the blade pitch angle based on the measured error in the angular speed (ω − ωrated). The blade pitch command βcmd is determined as:
βcmd = KP · (ω − ωrated) + KI ·
Z
(ω − ωrated) dt (2.14)
In the literature [22, 25, 30], the gains KP and KI are damped with a linear
gain scheduling in the blade pitch angle β: KP(β) = KP,0· βK βK+ β (2.15) KI(β) = KI,0· βK βK+ β (2.16) where Ki,0 is the base gain (i = P or i = I), and βK is the pitch angle
at which the pitch sensitivity (dP/dβ, where P is the rotor power) doubles respect to the design angle (β = 0):
dP dβ β=β K = 2 dP dβ β=0
¨ β = 1 τβ h Kβ· (βcmd− β) − ˙β i , (2.17)
where τβ is a characteristic time scale of the actuator, Kβ is a constant and β
is the current blade pitch angle. The values of constants in the previous equa-tions are taken from the reference values for the NREL 5-MW turbine [30]. The blade pitch command is saturated to 15◦
/s to prevent unstable dynam-ics in the Region III-Region II transition (the blade pitch is kept constant at β = 0, the design value, in Region II). The blade pitch β determined from equations (2.14)–(2.17) affects the aerodynamic torque by varying the design twist angle of the blade φ, equation (2.4). When the turbine is in Region III, the blade pitch is typically raised from the Region II zero angle value to a positive angle, thus increasing φ: this behavior is called pitch to feather. In this way, the controller reduces the angle of attack (Figure 2.4) and de-creases the aerodynamic torque, which allows to maintain an equilibrium at ω = ωrated.
2.3
Governing equations
We consider the flow over roughness to be incompressible. The governing equations are the continuity equation for incompressible flows and the Navier-Stokes Equations: ∇ · UUU = 0, (2.18a) ∂UUU ∂t + UUU · ∇UUU = − 1 ρ∇p + ν∇ 2UUU , (2.18b)
with UUU = (U, V, W ) is the velocity vector, p the pressure, ρ the fluid density and ν the kinematic viscosity. The equations (2.18) are adimen-sionalised by choosing reference length L and reference velocity U0. These
reference scales are arbitrary: tipically they are choosen as representative of a characteristic dimension and velocity of the problem under study. In our simulations on the rough wall, L is the height of half-channel measured starting from the plane of the crests on the rough wall, and U0 is the initial
condition on velocity, which is also the inlet condition at time 0.
As a result of the non-dimensionalization, we write the nondimensional Navier-Stokes equations.
∇∗· UUU ∗ = 0, (2.19a) ∂UUU∗ ∂t∗ + UUU ∗ · ∇∗UUU ∗ = −∇∗p ∗ + 1 Re∇∗ 2UUU∗ . (2.19b)
Asterisks denote non-dimensional variables: U UU∗ = UUU U0 t∗ = tL U0 p∗ = p ρU2 0 and: [∇∗]i = ∂ ∂x∗ i with x∗ i = xi L.
In the non-dimensional momentum equation (2.19b), the Reynolds num-ber Re is defined as Re = U0L
ν . It is the only similarity parameter in the
governing equations. According to the dynamic similarity principle [36], two flows geometrically similar with same Re, obey to the same equations (2.19), and they will have the same solution in terms of non-dimensional variables. For this reason, the non-dimensional approach is fundamental, and it is one of the main motivations of its wide use in Fluid Mechanics. The numerical code described in the following sections uses the non-dimensional form of the Navier-Stokes equations. For simplicity, the asterisks will be omitted and it is implied that all variables are appropriately normalized by the length and velocity scales.
2.4
Numerical Method
2.4.1 Discretization of Navier-Stokes equation
A numerical discretization is required to solve equations (2.18). For dis-cretizing in space, the numerical code used for this study applies central finite-difference method with an implicit discretization for linear terms and a third-order Runge-Kutta scheme for the non-linear terms.
The mesh-grid of the computational domain, as shown in figure (2.7), is a staggered orthogonal-cartesian grid, thus the velocity components are computed on the faces of the grid cells, while the pressure is the one in centre of the cell. This kind of grid implies a strong coupling between the pressure and the velocity components, which allows to avoid spurious pattern to arise in the pressure field [15]. It also has the advantage that discrete operators are
x y z U V W p
Figure 2.7: Grid cell with staggered velocities.
naturally expanded at the correct point when using central finite difference and this keeps the numerical scheme compact.
Linear terms
The discretization in time of the linear terms employs the Crank-Nicholson scheme referred to the instance t = tn+1/2 :
Uin+1− Uin ∆t + Ni(UUU n) = ∂p ∂xi t=tn+1/2 + 1 ReLjj Uin+1+ Uin 2 (2.20) where ∆t = tn+1−tnis the time step, ∂x∂i is the discrete gradient operator
computed with a centered finite-difference scheme and Ljj is the discrete
Laplacian operator. The term Ni(Uin) indicates the non-linear convective
term which is explicitly computed as a function of the solution Un
i at time
t = tn.
Non-Linear terms
The non-linear terms of the Navier-Stokes equations are extrapolated from the two previous time steps due to its easier implementation and because it allows avoiding linearization, which decreases the overall accuracy of the method. In particular, the scheme implemented is a low-storage third-order Runge-Kutta method. It was developed by A. Wray in 1987 [31, 55]. It advances from time t = tn to time t = tn+1 using three fractional steps and
guarantees third-order accuracy in time and also increases the stability of the numerical scheme. This method is defined low-storage, since it is requires the same amount of memory storage of a second-order scheme, while being third-order accurate.
At each intermediate step l the updated solution for the generic compo-nent Ui is computed as:
Uil+1 = Uil+ ρl∆t Ni(UUUl) + γl∆t Ni(UUUl−1) l = 0, . . . , 2 (2.21)
with Ul=0
i = Uin (the solution at time tn) and Uil=3 = Uin+1 (solution at
the next time step tn+1). The term N(UUU ) here generically indicates the
right-hand side of the equations. ∆t = tn+1− tn is the discrete time step.
The six coefficients ρi and γi can be determined comparing the previous
equation (2.21) with a Taylor expansion for Un+1
i , centered in Uin, up to the
third order, as shown in Orlandi (chapter 2, [55]).
The coefficient ρ0 is set equal to zero. In this way, no information is
needed from the previous time step with the exception of the starting solution uni. Several sets of coefficients may be employed, in the present code:
ρ0 = 0 ρ1 = − 17 60 ρ2= − 5 12 γ0 = 8 15 γ1 = 5 12 γ2 = 3 4 (2.22)
Solution and pressure correction
The discretized momentum balance equation is obtained by merging the dis-cretization of the linear and non-linear term, as a consequence the same time advancement must be applied to both the terms. This may be accomplished by centering the Crank-Nicholson scheme on l + 1/2 for each Runge-Kutta sub-step. Consequently, the discrete time step ∆tl is computed as:
∆tl= αl∆t (2.23)
where αl= ρl+γlensures the consistency of Crank-Nicholson with the
Runge-Kutta operator. Hence, it is possible to write the discretized Navier-Stokes equation as: Uil+1− Ul i ∆t +ρlNi(UUU l)+γ lNi(UUUl−1) = −αl δp δxi l+1 2 +αl 1 ReLjj Uil+1+ Ul i 2 !
n+1/2 n
In order to solve this issue, it is used a fractional step method firstly in-troduced in [32] to computational fluid dynamics applications and later im-proved by [49] to take account for generic boundary conditions. This method is one of those defined as projection methods, which are in general sub-divided into pressure-correction methods, velocity-correction methods and the consistent spitting methods. Guermond, Minev, and Shen [23] showed that the method [32] is equivalent to the incremental pressure-correction scheme in rotational form. The velocity-correction schemes are based on the idea of switching the roles of velocity and pressure with respect to pressure-correction methods. Moreover, the consistent splitting methods, introduced in [24] and equivalent to the gauge method [72], consist of computing the ve-locity and pressure in two steps: the first one computes the veve-locity treating the pressure explicitly, while the second is an update of pressure, based on the weak form of a Poisson equation for the pressure itself.
The method consists in solving equation (2.4.1) with the known pressure field at time t = tl, obtaining an intermediate non-solenoidal velocity field,
b
Ui. The velocity field bUi is then projected on a solenoidal space enforcing
the continuity equation and then used to update the pressure field pl+1.
In the following paragraphs, the overall solution and the pressure correc-tion will be reported. Substituting Ul+1
i with bUiin (2.4.1) allows to consider
the pressure at t = tl.
Introducing the new variable ∆Ui= bUi− Uil, one has:
∆Ui+ αl∆t Re Ljj ∆Ui 2 = − αl∆t δp δxi l − ρl∆t N (UUUl) − γl∆t N (UUUl−1) + αl∆t Re Ljj(U l i) (2.24) δjj+ αl∆t 2Re Ljj ∆Ui= Hi (2.25)
where Hi denotes all the known terms on the right-hand side, δjj is the
Kronecker’s delta.
Due to the stencil of the second-order discretization of the spatial deriva-tives (Ljj operator), the matrix on the left-hand side is a seven-diagonal
sparse matrix. Inversion of such matrix is too expensive on the computa-tional point of view for the typical dimensions of a numerical simulation: a
(coarse) computational grid may consist of N ∼ 106 points, and the cost
of a matrix inversion is in the order of N3 floating point operations. In
practice, this operation is never performed in an exact way. In our code, Equation (2.25) is solved with an approximate factorization technique in three steps: δjj+ αl∆t 2Re L1 1 ∆U∗∗ i = Hi δjj+ αl∆t 2Re L2 2 ∆U∗ i = ∆U ∗∗ i δjj+ αl∆t 2Re L3 3 ∆Ui= ∆Ui∗, (2.26)
where Lii is the discrete second derivative operator in the i-th direction.
The matrix resulting from the implicit terms involves only three non-zero diagonals for each direction i. The matrix can be inverted using the efficient Thomas’ algorithm, which costs just N in terms of floating point operations. This method is second-order accurate in time (∆t2).
A scalar quantity φ, called pseudo-pressure, is used to project bUi onto a
solenoidal space, hence the solution Ul+1
i is obtained from bUi = ∆Ui− Uil:
Uil+1= bUi− ∆t
δφ
δxi. (2.27)
A Poisson equation is solved in order to determine the scalar φ: it is de-rived from the divergence of equation (2.27) imposing the solenoidality of the velocity field at the new time step:
1 ∆t δ bUi δxi = δ δxi δφ δxi (2.28) The scalar φ is defined for consistency of the operators at the cell center and we can directly express its relation with pressure using (2.27) in (2.4.1):
pl+1= pl+ φ − αl∆t
2Re Ljjφ (2.29) At the boundaries, imposing the continuity of the new velocity field is equivalent to prescribe homogeneous Neumann conditions for φ. The Poisson equation with homogeneous Neumann boundary conditions is determined to within an additive constant. There are at least two ways to resolve the mathematical indefiniteness: the value of the pressure can be imposed at one point of the domain, or, just like in this code, the mean pressure can be
the pressure differences are significant for the flow, rather than the absolute value of the pressure.
Of course, a forcing term for the Navier-Stokes equations is always needed in order to sustain the motion of the flow inside the computational domain. Three choices to model this term are available: the constant flow rate strategy (CFR) is usually preferred since it is faster to adapt to a new state and shows
shorter transients. It consists of adjusting the pressure gradient Π = −∂p ∂x1 to keep constant the flow rate. Other possible choices are the constant pressure gradient (CPG) and the constant power input (CPI). As already mentioned, the pressure-correction procedure enforces the continuity equation, thus it is a CFR strategy. Anyway, Quadrio, Frohnapfel, and Hasegawa [61] stated that the choice of the forcing term does not produce important statistical consequences, unless one is interested in the strongest events of high wall friction.
We can recap our procedure for CFR as follow:
1. the intermediate non-solenoidal velocity field bUi is the solution of
equa-tion (2.4.1) without any forcing term;
2. a correction, induced by a forcing term, is introduced to recover the constant flow rate with (2.27);
3. the amplitude of the correction is computed solving by imposing the solenoidality (thus the continuity) of the new velocity field and solving the Poisson equation for the φ;
4. velocity and pressure at time level tl+1 are updated with (2.29)-(2.27)
and the next iteration is ready to begin.
In case of periodic boundary conditions both at inlet and in the spanwise direction, we impose the forcing term directly in 2.19b, then we integrate on the computational volume.
ZZZ V ∂U1 ∂t + ∂ ∂xj U1Uj dV = ZZZ V −∂x∂p 1 + 1 Re ∂2U1 ∂xj∂xj dV (2.30) The terms U1U1 and U1U3 of the second left hand side term vanish. The
remaining term U1U2 vanishes for the no slip conditions on the wall.
Equa-tion (2.30) can be written as: ∂Q ∂t = Π · V + 1 Re ZZZ V ∂2U 1 ∂xj∂xj dV (2.31)
where Q = RRR U1 dV is the variation of the total mass flow rate and
Π = −∂x∂p1 is the pressure gradient, which is uniform in the streamwise direction.
To keep constant the mass-flow rate, the applied pressure gradient is equal to: Π = −V1 Re1 ZZZ V ∂2U1 ∂xj∂xj dV (2.32)
It can be use as a parameter to find out whether the flow is in laminar or turbulent state. Clearly, assuming a steady value, it is much higher than one in laminar condition since more energy is dissipated by the turbulence. Stability and accuracy of the code
The numerical scheme is second-order accurate in space and time. The Runge-Kutta method used for the time-advancement is third-order accu-rate, but the higher precision is lost in the complete scheme. As already mentioned, the scheme is low-storage, meaning there is no additional burden from a memory viewpoint in using a third-order scheme for second-order accuracy.
The actual advantage of the Runge-Kutta scheme is its stability. Due to the explicit treatment of the convective terms, the CFL condition— named after Courant, Friedrichs and Lewy [12] —applies:
CFL = ui∆t ∆xi max ≤ CFLmax (2.33)
The CFLmax depends on the numerical scheme and its specific stencil. This
condition limits the maximum distance travelled by a particle in a time step respect to the mesh size. The CFL condition restrains the amplitude of the time-step, increasing the cost of a simulation as the CFL number is smaller (more steps are necessary to elapse the desire simulated time). This limitation has a central role in computational fluid dynamics since it does not allow to refine the mesh keeping larger time step to reduce computing cost: as the grid is refined, the CFL condition becomes more and more limiting and the cost increases. Anymway, Runge-Kutta method is multi-stage (it has three sub-steps), which allows to relax this condition, and one can afford a larger CFL number. While for most of the methods CFLmax .1, for the
three-step Runge-Kutta:
CFLmax=
√
act to stabilise the solution. This means that a time-step ∆t for which CFL > 1.7 is probably stable, although it is not possible to establish the actual limit a priori, since it depends on the solution. The implicit treatment of the convective term would permit to completely avoid the CFL limit. However, this would require linearization of the convective terms in each step, which is computationally very demanding.
Another general stability restriction is related to the viscous term [60]: ∆t
∆x2Re ≤
1
2n, (2.34)
where n expresses the dimensions of the problem (usually bidimensional or tridimensional problems). In practice, it means that information diffused by the viscous term ν∇2UUU shall not propagate for more than one cell-grid size in
one time step. This limit is very restrictive for tridimensional low-Reynolds flows. However, in this case the implicit treatment of the viscous term waives this limit, and stability is guaranteed only by the CFL condition.
2.4.2 Immersed Boundary Method
In the present work, bodies are modeled with Immersed Boundary Method (IBM): both waviness on the rough wall for the roughness simulations and tower with nacelle for wind farm simulations are simulated in this way. Mohd-Yusof [46] and Fadlun et al. [14] firstly developed this method, whose present implementation is due to Orlandi and Leonardi [56].
It consists mainly in imposing the velocity of the body to all grid points occupied by the body volume. In this particular case, all grid points that stay below the wall bound (black points of figure have therefore velocity equal to zero. One must pay attention when dealing with the boundaries of the wall. Without any correction, the body would be described in a step-wise way leading to inaccurate results. As a correction, the real contour of the wall is simulated by replacing the mesh distance with the effective distance from the real boundary of the body, when computing spatial derivatives that involves points across the body boundaries as shown in Figure 2.8.
2.4.3 Direct Numerical Simulation approach
In order for the numerical simulation to be accurate, the grid must be fine enough to include in the computation all the turbulent structures within the
Figure 2.8: Geometry with immersed boundary method.
flow. This means that grid size ∆x and the time step ∆t must be small enough to be able to capture the dynamics of the smallest scales of the flow. This kind of simulation is the so-called Direct Numerical Simulation (DNS) and it may become too challenging when a very large range of scales characterises the problem. According to classic energy cascade scheme [34, 62], the cascade of energy is the process in which the energy injected in the flow at larger scales is redistributed from the large eddies to the smaller ones, up to the smallest scales (i.e. eddies of a characteristic size and velocity). They are those whose energy is still large enough not to be dissipated by viscosity.
In the case of isotropic homogeneous turbulence, it is possible to quantify the ratio of characteristic size of the large eddy, L, to that of the smallest active one, η, a.k.a. Kolmogorov scale. In 1941, dimensional arguments lead A.N. Kolmogorov to this historical result:
L η ∝
Re3/4, (2.35)
where Re is the Reynolds number, based on the large-scale characteristic length and velocity: Re = UL/ν. Analogously, the ratio of the characteristic times of the eddies is:
T tη ∝
of the simulation if one wants to catch all the active scales. A well-resolved three-dimensional DNS should have a grid-spacing δ small enough to capture eddies of size η, and a computational domain range ∆ large enough to fit the largest scale L too. The scale ratio L/η becomes the ratio of the mesh range-to-spacing ∆/δ, which means the number of grid-points in one direction. Consequently, the Reynolds number provides an estimate for the number of grid-points, that is the unknowns, of a 3-D simulation: N ∝ Re9/4.
For the simulations on complex terrain, reported in the next chapter, a coarse DNS has been employed: the attribute coarse is due to the fact that not all the scales are resolved. Anyway, in a brief comparison, the results of simulations with and without model seem to be pretty similar.
2.4.4 Large-Eddy Simulation approach
In most of practical applications, such as aeronautical and wind farm sim-ulations, the Reynolds number is of the order of 107 − 108. For example,
modern utility scale wind turbine may have a diameter D as large as 100 m and a design wind speed U around 10 m/s), so the Re = UD/ν is of about 108. As a consequence, the number of points required for DNS would become
impossible to reach even for the most powerful modern supercomputers. This is the reason why alternative numerical approaches are fundamental in fluid dynamics: one of these is represented by Large-Eddy Simulations (LES). This method consists of an attempt at solving only the large scales of flow, which contain most of the energy and are directly linked to the particular problem. On the other hand, the smallest scales are not expected to be closely related to the particular flow, but are supposed to be universal. Therefore, they are modeled based on the resolved large scales and this reduces the size of the grid δ required to simulate the flow decreasing the overall number of points.
The universality of the small scale is implied in the local isotropy hypoth-esiswith the Kolmogorov theory. This scales, being small, are independent of the flow, and thus isotropic and statistically homogeneous. This assump-tion holds better as far as the Reynolds number is larger, since the scale separation increases (equation 2.35). Furthermore, the energy cascade pro-cess is long and it is reasonable to assume that the turbulent structures, as their scale decreases and the energy is transferred, lose memory of the large
anisotropic flow-dependent eddies and tends to return locally isotropic. A deep understanding of physical and theoretical framework is essential for the model. If the small scales are universal, so should be the model, or at least its underlying mechanisms.
2.4.5 Filtered Navier-Stokes Equations
The scale separation to identify the resolved scales is usually performed by applying a filtering operation to the Navier-Stokes equations. The filter is a mathematical tool which allows to remove all the scale lower than a selected cut-off scale (scale high-pass filter or, equivalently, frequency low-pass filter). It is defined as a convolution product between the physical variable φ(xxx, t) and the filter kernel G(xxx, t):
φ(xxx, t) = +∞ Z −∞ +∞ Z −∞ G(xxx − ξξξ, t − t′ )φ(ξξξ, t′ ) dt′ dξξξ (2.37)
where φ(xxx, t) is the filtered, resolved, variable. The unresolved part of φ(xxx, t) is denoted by a prime and is defined as:
φ′
(xxx, t) = φ(xxx, t) − φ(xxx, t) (2.38) The filter kernel is characterized by a cutoff length scale ∆ and a cutoff time scale τc. It requires some properties such as linearity, conservation of
constants and commutation with derivatives [63].
After application of the filter,the filtered Navier-Stokes equations are obtained: ∂Ui ∂xi = 0 (2.39a) ∂Ui ∂t + ∂ ∂xj UiUj= − ∂p ∂xi + 1 Re ∂2U i ∂xj∂xj +∂τij ∂xj (2.39b)
where τij is the so-called SubGrid-Scale (SGS) tensor, which represents the
interaction between the resolved scales and the unresolved ones which other-wise would not be present in the simulation. The subgrid tensor is defined as:
τij = UiUj − UiUj (2.40)
The appearance of this term in Equation (2.39b) is due to the non-linearity of the convective term and reflects the coupling between the large resolved
UiUj = Ui+ Ui′ Uj+ Uj′ = UiUj + UiUj′+ Ui′Uj+ Ui′Uj′ = UiUj + Cij+ Rij (2.41)
where Cij = UiU′j + U′iUj is the cross-stress tensor which represents the
interactions between the resolved and the subgrid scales. Rij = U′iU′j is
the SGS Reynolds tensor and represents the interactions among the subgrid modes. It should be possible, at least from a theoretical point of view, to evaluate all the terms which appear in the filtered momentum equation directly from the filtered field itself. Since the term UiUj would require a
second filtering, equations (2.41) can be further manipulated into:
UiUj = UiUj+ Lij+ Cij + Rij (2.42)
where Lij = UiUj − UiUj is the Leonard tensor, which represents the
inter-actions among the resolved scales. The subgrid tensor is composed of three contributions:
τij = Lij + Rij + Cij (2.43)
This is the Leonard triple decomposition, named after Leonard who first pro-posed this viewpoint in 1974 [38]. It provides a physical interpretation of the mechanisms which the SGS tensor accounts for. Other decompositions are possible based on different observations or definitions. For examples, the Leonard double decomposition [38], which arises from equation (2.41), or the Germano consistent decomposition [19], which is formally identical to equation (2.43) and can be seen as a generalization of the Leonard triple decomposition [63].
To solve the closure problem of the system of equations (2.39), the sub-grid tensor has to be modeled as function of the filtered velocity field UUU (closure problem). A widely used type of models are the eddy-viscosity
mod-els. These models describe the effects of the SGS terms analogously to the viscous mechanisms which take place at a molecular level in the fluids such as momentum or thermal exchanges.
In practice, the motion of the subgrid structures is assumed to be analo-gous to the Brownian motion of the molecules. The interactions which in the gas kinetic theory occur at microspic level, analogously in turbulent motions are assumed to occur at macroscopic level and to be relevant to the fluid
particle in opposition to the molecule. Subgrid, or eddy viscosity νsgs is also
introduced in analogy with the molecular case.
From Boussinesq’s hypothesis [17, 41], the deviatoric part1 of the SGS
tensor is expressed as a function of the symmetric part of the velocity gradient tensor, also called rate of strain tensor:
τd= −νsgs ∇UUU + ∇TUUU
(2.44) where the superscript ‘T ’ indicates the transpose operator. With defini-tion (2.44), the interacdefini-tions between the resolved and unresolved scales has the same layout as the viscous stresses:
ν∇2UUU = ∇ ·ν ∇UUU + ∇TUUU.
The main drawback of this formulation is that the subgrid tensor princi-pal axes are parallel to those of the strain rate tensor, but usually this does not happen in real flows [63]. Moreover, the resolved strain rate tensor is dominated by the large-scale structure and thus it is seldom isotropic as the small scale are expected to be.Other approaches are possible, such as the mixed models [1, 64, 75], gradient models [11, 42] and others (see [63] for a review).
We used the Smagorinsky model [67] in this study. It has been the first model advanced in the research community, and it is still the most used today. The subgrid viscosity is modelled as:
νsgs = Cs∆2
q 2SijSij
= Cs∆2 S
(2.45) where Sij is the rate of strain tensor and Cs is Smagorinsky constant, which
in general depends upon the type of flow and usually ranges between Cs =
0.1–0.2. In this code, Cs= 0.09 is used based on previous works [10].
The model is local and efficient, and it represents well the globally dissi-pative nature of turbulence. The subgrid dissipation provided by the model,
1
The decomposition is necessary since the strain rate tensor has zero trace. The isotropic part of the tensor 1
3τkkδijis lumped into the pressure term of the Navier-Stokes
equation. The filtered pressure p is replaced by the modified pressure p⋆: p⋆=p +1
3τkk
The subgrid contribution to the pressure may be significant and may be computed with the aid of some kind of estimate for the subgrid kinetic energy qsgs=12τkk
that locally the dissipation may be negative (back-scatter or backward energy cascade), but the model cannot catch this process.
The main drawback of the Smagorinsky model is its behaviour near to the wall, where the flow is always essentially laminar and the subgrid viscos-ity should vanish. On the contrary, with the definition in equation (2.45) the subgrid viscosity remains finite. A possible solution, also employed in the present code, is the use of a damping function, such as Van Driest’s damp-ing [69]. These functions artificially modify the constant Cs near to the wall
bringing it to zero. Van Driest’s damping is defined as: (Cs∆)2damped = h Cs∆ 1 − ey+/Ai2, (2.46) where y+ = u
τδ/ν, where δ is the distance from the wall and uτ = √τw
is the friction velocity (τw is the shear at the wall) and A = 25 is the
Van Driest’s constant. Damping functions are typically derived from classic smooth-wall turbulence analysis, and may not perform very well in pres-ence of complex geometries (where it is also difficult to properly estimated τw). In recent years, eddy-viscosity models have been developed for which
the subgrid viscosity vanishes at the wall, such as the σ-model [53], WALE (Wall-Adapting Local-Eddy) model [52] and the Vreman model [71]. Another alternative, is the use of the dynamic version of the Smagorinsky model [20]. In this version, Cs is not assumed as constant, but it is a scalar variable
computed based on the resolved field with the assumption of scale invari-ance. A second filter with a larger cut-off is used to modulate the energy transfer among scales on the basis of the resolved energy decay. This model significantly improves the shortcomings of the static (classic) Smagorinsky model, but also involves a quite larger computational cost because of the double (explicit) filtering.
In general, the filter operation (2.37) is a key process of the LES ap-proach, besides the definition of the sub-grid model. Though several filters can be built with the required properties [63], the explicit application of equa-tion (2.37) is seldom performed, mostly because of its high computaequa-tional expense. Instead, the filtering operation is meant to be implicitly applied by the computational grid. This procedure is completely inexpensive from a computational point of view and exploits the sampling effect due to Nyquist theorem. The grid is equivalent to the application of top-hat filter with cut-off frequency fc = 1/(2∆x), or cut-off wavenumber kc = π/∆x, and thus all
The choice of the mesh size has to ensure that the cut-off includes all the dynamically active scales of the flow, i.e. the cut-off wavenumber has to be in the dissipative range of the energy spectrum of the solution. Of course, this size ∆x is not known a priori and that is a potential source of error in the numerical simulation. Usually, grid independence studies are performed to check the convergence of the solution.
Another common problem which may arise especially when simulating complex geometries is related to the topology of the grid. If the grid, or the filter, is not uniform then commutation with the derivatives is not guaranteed and this generates a commutation error. In this case the error should be eliminated or controlled, for example by employing high-order commuting, explicit, filters [70].
Simulations over complex
terrain
A parametric study varying the wavelength of the rough walls has been per-formed to characterize the flow over a complex terrain. For this reason, a three-dimensional rough wall seems to be more realistic. This fact must not be confused with the intention of reproducing the atmospheric bound-ary layer accurately, even if it takes inspiration from that. In practice, the roughness consists of a sinusoidal three-dimensional waviness obtained by multiplying one sinus in spanwise and the other in streamwise direction, as follow: y = a sin 2π λx x sin 2π λz z + 1 (3.1) where a is the amplitude of the sinusoidal, λx and λz are respectively the
streamwise and spanwise wavelength of the geometry. Tipically, they are referred as λx/a and λz/a, the aspect ratio of the complex terrain. From now,
this notation will be used, so that we can easily compare these simulations with some others in literature, even with slightly different geometries. The origin of the wall normal coordinate y is at the plane of the crests for sake of simplicity. One of the first studies with roughness similar to the one presented is due to Bhaganagar, Kim, and Coleman [2], which deal with a roughness similar to an egg cartoon.
3.1
Simulation setup
The computational domain is an half-channel with the rough wall at the bottom and its dimensions can be found in Table 3.1, where Lx and Lz
are the dimensions in streamwise and spanwise directions respectively, Ly in
the wall normal direction. All the dimensions are non-dimensionalized with respect to the reference length, the height of the half-channel measured from the plane of the crests δ.
Case ID λx 2a λ2az Lx Ly Lz Lλxx Lz λz 1.1 6 6.4 6 1.1 3.84 10 6 1.2 6 12.8 6 1.1 3.84 10 3 2.1 12 6.4 6 1.1 3.84 5 6 2.2 12 12.8 6 1.1 3.84 5 3 3.1 24 6.4 7.2 1.1 3.84 3 6 3.2 24 12.8 7.2 1.1 3.84 3 3
Table 3.1: Setup parameters
The nodes of the grid are 480×192×320 respectively in x, y, z directions. Spacing of the grid is uniform in streamwise and spanwise directions. Instead, in the wall-normal direction it is constant near to the wall and then stretched, as shown in Figure 3.1.
In Figure 3.1(a), the y coordinate of each grid point is plotted with respect to the corresponding node (node 0 is at the bottom of roughness). Next to that, in Figure 3.1(b) it can be observed how grid spacing varies along the y direction. Dashed lines mark the position of plane of the crests. The resolution is kept costant and maximum, thus with minimum spacing ∆y = 0.004δ up to y∗ = 0.2; then it is gradually reduced, reaching its
minimum resolution (with ∆y = 0.008δ) at the top boundary.
(a) 0 0.2 0.4 0.6 0 20 40 60 80 100 120 140 160 180 y node y /δ (b) 0 0.002 0.004 0.006 0 0.2 0.4 0.6 0.8 1 y/δ ∆ y
Figure 3.1: (a) Grid y-coordinates (b) Grid spacing in y-direction
following transformation: yi = y0+(i−1)yN∗ ∗ y y ≤ y∗ tanh α Ny −N∗i−1 y −1 2)) tanh α 2) y > y∗ (3.2) where Ny and Ny∗ are the wall-normal grid points and the number of points
with constant spacing, y0 and y∗ are the origin of the y-coordinate and the
final point with constant spacing, α is a scaling parameter. This resolution allows to maintain a finer grid where the gradients are higher, which is also where the most relevant phenomena of turbulence take place.
The geometry of the complex terrain has fixed amplitude, a = 0.05δ for all the simulations, thus the height of peak is equal to 0.1δ(= 2a). Each sim-ulation is characterized by a different couple of wavelengths [λx/2a, λz/2a]:
in particular with three streamwise and two spanwise, thus six cases are con-sidered, as shown in Table 3.1. Note that Case ID is in the format X.X, where the two numbers refers the streamwise and spanwise wavelengths re-spectively. Higher numbers imply longer wavelengths.
Periodic boundary conditions are applied in both streamwise and span-wise directions. No-slip condition is imposed at the bottom boundary, whereas at the top free slip condition is used.
In this chapter, we will discuss the main results of simulations over com-plex terrain: in the first part, the simulations at low Reynolds number are presented. They have been performed with the aim of testing the geometry and validating results with the existing literature. Then, other simulations at high Reynolds number are reported: these are basically employed as pre-cursors for wind farm simulations.
3.2
Coarse DNS Simulations at Low Re number
In principle, the DNS should solve all the scales of turbulence within the flow. As a consequence, the choice of grid spacing should be based of the size of the smallest turbulent structures. According to Kolmogorov’s theory [34], the characteristic length of the smallest scales, η, also known as Kolmogorov’s scales, depends on the kinematic viscosity of the fluid, ν, and on the viscous dissipation rate of energy, ǫ = ν|∇u2|:
η = ν3 ǫ 1/4 (3.3) Considering a statistically steady motion, equilibrium must be guaranteed: the rate of energy introduced in the large scales from the mean field has to be equal to the transport rate of energy along the cascade of energy and to the dissipation rate in the Kolmogorov’s scales. Equation 3.3 can be rewritten as:
L η = Re
3/4 (3.4)
Where L is the reference length, in this case L = δ and Re is the Reynolds number. Consequently, the Kolmogorov’s scales η can be derived. Thus, the resolution should be equal to η near to the wall, but the computational cost would be unfeasible: since we perform simulations at Re = 104 and the
cost of simulations increases as Re9/4. The actual resolution in wall units
and the Reτ for each case are reported in Table 3.2. For this reason, these
simulation are more properly coarse DNSs since not all the turbulent scales are properly resolved, but we are neglecting part of them (the resolution is about 4η). Differently from Large Eddy Simulations, the subgrid scales are not even modeled in this case. This choice comes after solving the same simulation with both the LES and the coarse DNS, since the results are practically the same.
Simulations last 400 time units and for each case 200 flow fields are stored.
3.2.1 Results
In the first part, the results are not only averaged in time, but also in span-wise and streamspan-wise direction. Even if the hypothesis of planar homogeneity is an approximation for these simulations and holds only above the rough-ness layer (about 5 times the amplitude of the wavirough-ness), this assumption allows to consider the general behaviour of the flow since it is not the pur-pouse of this study to evaluate the local behaviour of the flow over rough
3.2 − 6.4 1.2 6 12.8 811 10.0 3.2 − 6.3 9.6 2.1 12 6.4 625 7.8 2.5 − 4.9 7.5 2.2 12 12.8 663 8.2 2.6 − 5.2 7.9 3.1 24 6.4 555 8.3 2.2 − 4.4 6.7 3.2 24 12.8 550 8.3 2.2 − 4.4 6.6
Table 3.2: Low-Re simulation: Reynolds number and resolution in wall units.
wall. Chan et al. [7] carried out a similar analysis on a turbulent pipe flow with wavy three-dimensional wall using the triple decomposition of the ve-locity field, which is arguable, but it is still present also in this work due to the necessity of applying space average. In the next pages, the results of simulations will be compared in order to assess what are the difference related to the variation in the roughness wavelengths. Bhaganagar, Kim, and Coleman [2] performed simulations on a plane-channel flow between a smooth wall and one covered with regular three-dimensional roughness el-ements, named egg-cartoon. Similarly to the present work, they compare results from simulations with different wavelength of the geometry in the streamwise and spanwise directions to isolate the effects on turbulent statis-tics and the instantaneous turbulence structure. In the next pages, we are going to compare our results in terms of the roughness function and the fluctuations of velocity with those in [2] to find out if and why they show some differences. Moreover, the results will be also compared with [13], which simulated a two dimensional and three-dimensional sinusoidal rough-ness of varying roughrough-ness height and wavelength. They discovered that the streaks are modified selectively by the roughness, since they are completely different between the two-dimensional and the three-dimensional case. In fact, the two-dimensional roughness breaks the streamwise structures since the flow cannot go around the roughness element as it happens for a three-dimensional roughness, whose structures indeed are longer. For this reason, it is necessary to use a three-dimensional in order to represent a realistic complex terrain. Differently from [13], our roughness has fixed amplitude similarly to [2]. As already mentioned the hypothesis of planar homogeneity is in principle not valid, thus the time average is the only really reasonable. As a consequence, we will compare the rms profiles obtained using the planar average with those in [7], then we verify these results with the rms time and phase averaged on the planes XY and YZ. Results from [49] are also shown as a reference for underlining the differences of mean velocity profiles with
respect to the smooth wall.
Each of the previously mentioned studies [2, 7, 13] has some different fea-tures with respect to this work, thus they allow to determine the reliability of our results, which is fundamental since we want to use the simulations at higher Reynolds number as precursor for the wind farm simulations. Further-more, this comparison is also useful to determine if the triple decomposition of the velocity field (which will be later introduced) can give some useful in-formations about the fluctuations of velocity even if in principle it is applied only to mitigated the effects of the space average, which is not valid due to the inhomogeneities induced by the roughness.
Mean velocity profile
Averaging the streamwise velocity in time and space, mean velocity profile are obtained. The velocity is non-dimensionalized with the friction velocity Uτ.
The space average of the streamwise velocity is performed in two different ways: in Figure 3.2 (a), it is averaged on all the grid points of each plane parallel to the top and bottom boundary, thus both on the points within the flow than on those inside the body, where the velocity is null. On the contrary, Figure 3.2(b) shows the mean profile calculated only on the points within the flow. This method represents better the flow within the cavities and the differences among the cases. In particular, it allows to notice in Figure 3.2(b) that the velocity is negative below the half of roughness height for case 1.X (the cases with the same λz are superimposed) probably due
to recirculations within the canopies of the roughness. The negative portion of the profile is reduced by the increase in streamwise wavelength of the geometry due to the decrease in the dimension of recirculations. De Marchis, Milici, and Napoli [13] noticed the same effect on the mean velocity profiles of 2D and 3D roughness. Considering the irregularity of their roughness, we need to compare the results in terms of the effective slope ES. We also should take into account the different procedure in computing the profiles, since they only average the velocity in time and not in space. Anyway, the region below the half height of the roughness shows recirculations whose dimension decreases for decreasing ES, as is shown in [13]. The amplitude of the peak of negative velocity for the profiles in both the studies is of about 0.5 − 1.0Uτ. The main differences with the profiles in [13] can be found in
the region between the half of the roughness height (y = −0.05) up to about y = 0.1, since in this region the space average lacks of validity due to local inhomogeneities, thus our profiles give only a general description of the flow.
near to the rough wall. This effect is linked to the different drag of one case with respect to the other. The Cd of each waviness is easily computed from
the pressure gradient applied to the simulation in order to sustain the motion of the flow with the periodic boundary conditions, as described in one of the following paragraphs.
As we consider longer streamwise wavelength, the velocity profile fits better the profile of the channel flow by [33] at Reτ = 395, this result is obviously
expected since a smooth wall can be considered as the limiting case with λx/2a and λz/2a going to infinite.
(a) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 y /δ U+ (b) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 y /δ U+
Figure 3.2: Mean Velocity Profiles: case 1.1 ( ), case 1.2 ( ), case 2.1
( ), case 2.2 ( ), case 3.1 ( ), case 3.2 ( ); KMM at Reτ= 395
[33]
Drag Coefficient
The Drag Coefficient, Cd, is the drag force per unit area that acts on the
rough wall. In the channel flow, the drag coefficient has only the contribute due to friction, but in this case, also pressure drag has an important role. It is usually defined as:
Cd=
D
ρU2S (3.5)
where D is the drag in streamwise direction, U is the reference velocity and ρ the density of the fluid. The reference surface S is choosen as the surface equivalent to the flat wall within the same computational domain, thus S = L1L3.
Once the flow is statistically steady, Cd can be easily computed with an
integral momentum balance. The applied pressure gradient has to balance the momentum deficit due to the rough wall, in particular due to the already
![Figure 2.1: Nikuradse’s Diagram [54]](https://thumb-eu.123doks.com/thumbv2/123dokorg/7503546.104659/22.892.231.667.191.477/figure-nikuradse-s-diagram.webp)
![Figure 2.2: Moody’s Diagram [47]](https://thumb-eu.123doks.com/thumbv2/123dokorg/7503546.104659/23.892.234.662.196.478/figure-moody-s-diagram.webp)
![Figure 2.3: Roughness Function. Adapted by Jiménez [28].](https://thumb-eu.123doks.com/thumbv2/123dokorg/7503546.104659/24.892.257.635.187.504/figure-roughness-function-adapted-by-jiménez.webp)
