4. Turbulence Inflow¶
4.1. Introduction¶
In computational wind engineering (CWE), the generation of inflow turbulence satisfying prescribed mean-velocity profiles, turbulence spectra, and spatial and temporal correlations is of great importance for the accurate evaluation of wind effects on buildings and structures. More specifically, the task is to generate a turbulent velocity field
where
should be spatially and temporally correlated.item
needs to have prescribed Reynolds stresses tensor where ( ) is the -th component of and the over line denotes the time average. needs to have prescribed integral length scales should fulfil the divergence free constraint . should have prescribed correlation functions or spectra.
Several methodologies have been proposed for this purpose which can be classified into three general categories: precursor simulation methods, recycling methods and synthetic methods. Compared with precursor simulation and recycling methods, synthetic methods in general offer a more practical and relatively efficient approach to generate inflow turbulence, and is therefore chosen as the subject of this section. Research activities on synthetic turbulence generation have been vigorous over the past decades and have branched out into several categories of techniques [Wu17], including the synthetic random Fourier method [Kra70, Hos72, MB21], the synthetic digital filtering method [KSJ03] and the synthetic eddy methods [JBLP06]. A brief introduction regarding to these techniques is given below and emphasis is placed on their abilities to capture the statistical characteristics as well as the spatial and temporal coherence of turbulence. Also note that since real turbulence is very complex, in most cases, not all of the above listed features can be fulfilled. There is always some adaptation time required for artificial turbulence to evolve into real turbulence. Fulfilling the properties above with the synthetic turbulence is important to minimize the adaptation time or length.
4.2. Synthetic Random Fourier Method¶
The so-called synthetic random Fourier method (SRFM) attempts to model turbulent flow field indirectly by imposing constraints on uncorrelated random fields through an energy spectrum to account for the spatial and temporal correlations, which can be further classified into two groups. The first group of the SRFM was based on the pioneering work in [Hos72] and [SJ72] on the simulation of multi-correlated random processes using a weighted amplitude wave superposition (WAWS) method. This approach has an advantage that both the targeted power- and cross-spectra can be imposed in the generation process so that the prescribed target characteristics can be maintained. A major drawback of this method is that the generated turbulence does not satisfy the continuity equation of the flow, or in other words, the divergence-free condition is not guaranteed. As a consequence, it would take enormous effort for the solver to enforce the continuity by correcting the turbulence inflow inserted into the computational domain, and the statistical characteristics of the corrected flow field differ from the target values. More recently, [MB21] proposed a computationally efficient implementation of the method in [SJ72] to generate inflow turbulence by enforcing continuity requirements posteriorly.
The second group of the SRFM was initiated by the work in [Kra70] on divergence-free homogeneous isotropic turbulence synthesis through the superposition of random harmonic functions. [SSC01] took a step forward by combining Kraichnan’s technique with scaling and orthogonal transformation operations in a procedure known as the random flow generation (RFG) which allows to generate inhomogeneous and anisotropic turbulence. However, the scaling operation introduced in the RFG technique can result in a velocity field that is not divergence-free for inhomogeneous turbulence. Modifications to enforce the divergence-free the constraint for inhomogeneous turbulence was discussed in [YB14]. A major drawback of RFG technique is that the power spectra of the generated turbulence only follow Gaussian spectra model, so it is not suitable for simulating flows in the atmospheric boundary layer. [HLW10] revisited Kraichnan’s method and proposed a technique called DSRFG (for discretizing and synthesizing random flow generation) which allows to generate turbulent inflow from any prescribed spectrum. Instead of using the scaling and orthogonal transformation, the anisotropy of turbulence is realized by modifying the distribution strategy of the wave vector in Kraichnan’s original method. A drawback of the DSRFG technique is that it produces fluctuating velocities with high correlation due to the fact that in this method the spatial correlation is modeled by a parameter which is not a function of frequency but a constant value. Inspired by the DSRFG method, [CPMS17] proposed some modifications to this technique to obtain the velocity field that had a better match with the target turbulent statistics. This method, known as modified discretizing and synthesizing random flow generation (MDSRFG), is capable of removing the dependence of statistical quantities of synthetic turbulence on spectra discretization resolution. [AEBED15] also proposed a technique called consistent discrete RFG (CDRFG) to accurately model the target spectra and the coherence function. In both two methods mentioned above, the parameter that characterizes the spatial correlation is expressed as a function of frequency to account for the damping of coherence with the increase of frequency. An attractive feature of second group of SRFM is that the generation procedures are usually independent at each point and each time-instant so that it can be easily accelerated by conducting parallel computation, although the generated random flow may not satisfy the continuity equation.
4.3. Synthetic Eddy Method¶
The synthetic eddy method (SEM) initiated by [JBLP06] is based on the classical view of turbulence as a superposition of the representative coherent eddies. In the SEM, the flow is assumed to consist of randomly distributed turbulent spots, and each turbulent spot is modeled by a three-dimensional shape function with compact support and satisfies a proper normalization condition. The spots are then assumed to be convected through an inlet plane with a reference velocity using Taylor’s frozen turbulence hypothesis. The resulting inflow turbulence is then reconstructed using the method proposed to recover the desired statistical characteristics and to account for the conditions of inhomogeneity and anisotropy. The choice of the shape function plays an important role in the SEM since it is directly related to the two-point autocorrelation function, and consequently the power spectrum of the synthetic turbulence. Enforcement of the continuity condition in the SEM was discussed in [PCR13].
A brief introduction on the SEM presented by [JBLP06] is given as follows. To start with, the turbulent spot mentioned above can be represented as eddies defined by shape function
The inflow plane on which we want to generate the synthetic turbulence with the SEM is basically a finite set of points
where
The volume of the box of eddies is noted by
where
The coefficient
where
where
with a new random intensity vector
Mean flow and Reynolds stresses¶
The mean value of the velocity signal (4.3.4) can be expressed as
where the angles denote the mean operator. The independence between the random variables
The term
The Reynolds stresses
Using again the independence between the random variables
The term
follows from the fact that
Finally, we arrive at
Hence the Reynolds stresses of the velocity fluctuations generated by the SEM reproduce exactly the input Reynolds stresses.
Two-point correlation¶
The two-point cross-correlation of the velocity fluctuations writes
where
Using again the independence between the positions
By (4.3.15), the term in the mean operator writes
Inserting (4.3.20) back to (4.3.19) and using (4.3.6), this yields
where
Recall the integral length scale
Fourier analysis can also be used to obtain the spectra of the synthetic turbulence. Note that
the velocity spectrum tensor
Recall the convolution theorem for cross-correlation states that
Hence the spatial velocity spectrum tensor can be expressed as
where
Two-time correlation¶
The two-time correlation tensor of the velocity, denoted by
By (4.3.4) and the linearity of the statistical mean, we have
The independence between the position
Consequently (4.3.28) reduces to
Before computing the term in the angles, we define
If
Since
In the case where the mean velocity is in the x-direction only
Thus the two-time correlation of the signal at time
Using again the convolution theorem as expressed, the above expression simplifies to
Commonly used velocity shape functions¶
We list three commonly used velocity shape functions
Tent function
Step function
Truncated Gaussian function
4.4. Digital filtering method¶
The synthetic digital filtering method (SDFM) initiated by [KSJ03] attempts to model the spatial and temporal coherence of turbulent inflow through the digital filtering uncorrelated random data, and account for inhomogeneity and anisotropy using the method proposed by [LWS98]. It is relatively easy to implement and is able to reproduce the first and second order one-point statistics as well as autocorrelation function. However, the synthetic turbulence generated by SDFM does not satisfy the continuity equation. [KCX13] offered a promising approach to enforce the divergence-free constraint in the SDFM by inserting the synthetic turbulence on a transverse plane near the inlet and relying on pressure-velocity coupling to do the correction. From a computational wind engineering point of view, the ability of SDFM to impose a two-point spatial correlation directly is very attractive.
We now briefly introduce the filtering method by [KSJ03]. In order to create two-point
correlations, let
defines a convolution or a digital linear non-recursive filter. The
where
For a general target correlation function, the filter coefficients
where
where
The width
It is suggested [XC08] to evaluate the filter coefficients using
where
Again, the width
An algorithm for generating inflow data may look like this (alternatively one can generate a large volume of data, store it and convect it through the inflow plane by applying Taylor’s hypothesis):
Choose for each coordinate direction corresponding to the inflow plane a length scale
, , a time scale and determine the filter width ( ) accordingly.Initialize and store three random fields
(again ) of dimensions where denotes the dimensions of computational gird of the inflow plane.
(c) Compute the filter coefficients
Applying the following filter operation for
,(4.4.10)¶which yields the two-dimensional arrays of spatially correlated data
, .Output velocity data with the transformation
(4.4.11)¶where the coefficients
are given by (4.3.5). This step ensures the synthetic velocity reproduces the target mean velocity and Reynolds stress tensor.Discard the first
-plane of and shift the whole data: . Fill the plane with new random numbers.Repeat the steps (d)
(g) for each time step.
If the target correlation function is an exponential function, an alternative approach by [XC08] can be adopted for generating inflow turbulence which turns out to be much more efficient than the method of [KSJ03]. Instead of using the filtering operation discussed above, Xie and Castro’s method obtain the temporal correlation with the expression
where
which reproduces an exponential function. An overall algorithm for generating the inflow velocity supported by the method of [XC08] can be stated as follows
Choose for each coordinate direction corresponding to the inflow plane a length scale
, , a time scale and determine the filter width accordingly.Initialize and store three random fields
(again ) of dimensions where denotes the dimensions of computational gird in the inflow plane.Compute the filter coefficients
with a prescribed function or by a multidimensional Newton method such that the resulting correlation function meet the target one.Applying the following filter operations for
,(4.4.14)¶which yields the two-dimensional arrays of spatially correlated data
, .Compute
with (4.4.12) and output the velocity signal with the transformation(4.4.15)¶where the coefficients
are given by (4.3.5). Again, this step ensures the synthetic velocity reproduces the target mean velocity and Reynolds stress tensor.Repeat the steps (d)
(f) for each time step.
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