**2. Aims and Methodology of This Review**

In this review, we aim to provide a comprehensive picture of the current knowledge of the role of lncRNAs in the pathogenesis of endometriosis and the mechanisms by which they affect the disease. We included patient studies to detect lncRNAs that have been associated with endometriosis, along with in vitro studies and in vivo animal studies to analyze the mechanism of action and function of lncRNAs in the disease. To gather studies for this review, we searched both the PubMed and Google Scholar databases for endometriosis and lncRNA combined with the following keywords: mechanism of action, sponging, ceRNA networks, chromatin, biomarkers, genome-wide, microenvironment, and infertility. We concentrated on original research papers, excluding literature reviews, pre-prints, meeting abstracts, and articles that were not in English and then assessed the remaining publications for this review.


**Author details**

Raquel Del Angel, Juan C. Durán-Álvarez\* and Rodolfo Zanella

\*Address all correspondence to<EMAIL_ADDRESS>
Institute of Applied Science and Technology, National Autonomous University of Mexico, Mexico City, Mexico

[11] Fagan R, McCormack DE, Dionysiou DD, Pillai SC. A review of solar and visible light

[13] Sclafani A, Herrmann JM. Comparison of the photoelectronic and photocatalytic activities of various anatase and rutile forms of titania in pure liquid organic phases and in aqueous solutions. The Journal of Physical Chemistry. 1996;**100**:13655-13661. DOI:

[14] Priecel P, Adekunle Salami H, Padilla RH, Zhong Z, Lopez-Sanchez JA. Anisotropic gold nanoparticles: Preparation and applications in catalysis. Chinese Journal of Catalysis.

cations: A review. Industrial and Engineering Chemistry Research. 2013;**52**:3581-3599.

ronmental applications. Journal of Photochemistry and Photobiology C: Photochemistry

Selected results and related mechanisms on interfacial charge carrier transfer dynamics.

Journal of Physical Chemistry A. 2011;**115**:13211-13241. DOI: 10.1021/jp204364a

[18] Marschall R. Semiconductor composites: Strategies for enhancing charge carrier separation to improve photocatalytic activity. Advanced Functional Materials. 2014;**24**:2421-

[19] Zhang Z, Yates JT. Band bending in semiconductors: Chemical and physical consequences at surfaces and interfaces. Chemical Reviews. 2012;**112**:5520-5551. DOI: 10.1021/

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2440. DOI: 10.1002/adfm.201303214

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cr3000626

for H2


**4. Discussion**

This study analyses the impact that performing ultrasound scans in primary care centres has in terms of avoided journeys to specialized services, time saved, reduced fuel consumption and the consequent decrease in the emission of atmospheric pollutants. The rural and dispersed nature of most of the study centres highlights a greater effectiveness of the intervention in this field. The county in which the study was conducted has a population of 158,334 (160,464 registered residents), of which 54.89% are assigned to a rural health area with under 10,000 inhabitants and less than 150 inhabitants/km2. From the patient's point of view, avoiding an unnecessary journey to the radiology service means, in addition to saving time, a reduction in fuel costs, while the same examination is performed in a manner which is similar to their own GP (primary care level).

This study joins others which show that when PCCs provide tests or services which until now have been offered exclusively by specialized units, it leads to a reduction in patient journeys and thus contributes to a significant reduction in the associated costs for users [32,33] in addition to a reduction in the emission of atmospheric pollutants [6,34]. According to data from the World Health Organization (WHO), air pollution in cities and rural areas is responsible for 4.2 million premature deaths per year due to exposure to particulate matter (PM) which causes cardiovascular and respiratory diseases and cancers [35]. Therefore, the reduction in emissions associated with the journeys shown to have been avoided would have a beneficial effect on the health of the population. In fact, the WHO has recently determined the environmental burden of diseases in each country based on selected risk factors including air pollution, and in the case of Spain the air pollution burden has been estimated to stand at 5800 deaths per year. This calculation assumes a reduction in the average urban levels of PM10 from 30 g/m3 to 20 g/m3, the average value of PM10 in the WHO's recent recommendations [36].

All factors in this study have been based on the mandatory reporting factors described in the EMEP/EEA air pollutant emissions inventory of the European Environment Agency [37]. Although carbon dioxide is a major factor in environmental pollution, it is not included in the inventory and has not been taken into account in the study as its inclusion could have altered reliability with respect to the calculations made with other factors that are in the EMEP/EEA inventory.

It should be noted that to facilitate the calculations made in the study and extrapolating from the most frequent case in the setting in which this study was conducted, the transport system used to make the journeys was taken to be a private, small family car with average fuel consumption. This study did not take into account any journeys which may have been made by public transport (very scarce or non-existent in certain rural populations), low or zero emission vehicles (hybrid and electric cars), or highly polluting vehicles (obsolete cars or towing a trailer).

Nonetheless, the present study has certain limitations. One being the fact that it is a retrospective study which does not allow us to obtain any information of interest such as data related to the loss of working hours and wages, the amount of stress related to driving, waiting times or additional costs such as parking charges. None of these costs are reflected in the current study. In addition, this study does not take into account factors which increase the cost of performing ultrasound scans in PCCs, such as the need to purchase equipment, the ongoing training of professionals, the additional time the professional needs to dedicate to conducting examinations or the possible increase in demand for ultrasound scans due to greater access to the test. The POCUS model could also lead to overdiagnosis if its use is not limited to the organs upon which the clinical suspicion that motivates the use is based.

One factor not assessed in the study, but which is of great interest, is the environmental impact generated by equipment wastage and consumable materials that might cause air and land pollution. Although this impact is not analysed in this study, we should also consider it in order to have a complete picture of the environmental impact. According to the purchasing department, the disposable materials relating to ultrasounds include gloves, disinfectant liquid, reel hand paper, a bunk bed, transducer covers and conductive gel. Among these materials, the only ones exclusive to ultrasounds are the conductive gel and the transducer covers (the rest are difficult to impute directly to the use of ultrasounds) [38]. The annual purchase of conductive gel for the radiology service was 713 units compared to 107 in primary care; 7776 transducer covers were purchased in the radiology service compared to 144 in primary care. Although the environmental impact of these wastes may be limited, it seems that there is a greater use of consumables and therefore a greater production of waste in the specialized services.

The study could also have evaluated other factors which a ffect the use of the POCUS model in primary care which go beyond the factors under investigation. Ultrasound scans performed using this model mean that any potential referrals can be optimised, minimizing uncertainty and ruling out certain diseases due to the equipment's high diagnostic precision. Ultrasound is an additional tool in the diagnosis process, but its use should be limited to certain clinical situations. Its use in the early detection of diseases prevalent in primary care ought to be appropriately evaluated [8].


**Author details**

Disha N. Moholkar† , Darshana V. Havaldar† , Rachana S. Potadar and Kiran D. Pawar\* School of Nanoscience and Biotechnology, Shivaji University, Kolhapur, India

\*Address all correspondence to<EMAIL_ADDRESS><EMAIL_ADDRESS>
† These authors contributed equally.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[12] Ceylan H, Giltinan J, Kozielski K, Sitti M. Mobile microrobots for bioengineering applications. Lab on a Chip. 2017;**17**:1705-1724 https://doi.

[13] Ornes S. Medical microrobots have potential in surgery, therapy, imaging. and diagnostics, Proc. Natl. Acad. Sci. U. S. A. 2017;**114**:12356-12358 https://doi.

[14] Bratlie KM, York RL, Invernale MA, Langer RL, Anderson DG. Materials for diabetes therapeutics. Advanced Healthcare Materials. 2012;**1**:267- 284 https://doi.org/10.1002/

[15] Ravaine V, Ancla C, Catargi B. Chemically controlled closed-loop insulin delivery. Journal of Controlled Release. 2008;**132**:2-11 https://doi. org/10.1016/j.jconrel.2008.08.009

[16] Thomas R, Park IK, Jeong YY. Magnetic iron oxide nanoparticles for multimodal imaging and therapy

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*Article* **A Family of Single-Stage, Buck-Boost Inverters for Photovoltaic Applications**

#### **Ben Zhao 1,\*, Alexander Abramovitz 2, Chang Liu 1, Yongheng Yang 3 and Yigeng Huangfu 1**

Received: 5 February 2020; Accepted: 21 March 2020; Published: 3 April 2020

**Abstract:** This paper introduces a family of single-stage buck-boost DC/AC inverters for photovoltaic (PV) applications. The high-gain feature was attained by applying a multi-winding tapped inductor, and thus, the proposed topologies can generate a grid-level AC output voltage without using additional high step-up stages. The proposed topologies had a low component count and consisted of a single magnetic device and three or four power switches. Moreover, the switches were assembled in a push-pull or half/full-bridge arrangement, which allowed using commercial low-cost driver-integrated circuits. In this paper, the operation principle and comparison of the proposed topologies are presented. The feasibility of the proposed topologies was verified by simulations and experimental tests.

**Keywords:** PV microinverters; converter topologies; single-stage; buck-boost; tapped inductor


3. Application of the finite element method to fluids

Due to the assumptions Eq. (12), only vr, vz velocity components and pressure p are chosen as primitive variables.

All stress variables are expressed in terms of the velocities and the pressures and the stresses are then back-calculated after finding the velocities vr, vz and the pressure p.

For stability the pressure field must be interpolated with a polynomial one order lower than the velocity terms. Here we have chosen linear pressure and quadratic velocity fields over the element.

A quadratic triangle for velocities with six nodes, three on the vertices and the three others on mid-sides, and a linear triangle for pressures, both processed with area coordinates (Figure 5).

In our problem gravitational and inertial forces are neglected and the motion is obtained as in the first Stokes problem by drawing the cylinder opposite to the sphere displacement with a constant, upward Vs velocity. To solve the nonlinear equations iterative optimization techniques are used.

#### 3.1. Linear and quadratic triangles

The Ne f g shape functions for velocities corresponding to the three vertice nodes and three midside nodes (1, 2, …, 6) are respectively using the well-known area coordinates L1, L2, L3.

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model http://dx.doi.org/10.5772/intechopen.75977 73

Figure 5. (a) A linear triangular element; and (b) a quadratic triangular element.

$$\rm L\_1(2\ L\_{1-1});\rm L\_2\ (2\ L\_{2-1});\rm L\_3(2\ L\_{3-1});\rm 4L\_1\newline L\_2;4\ L\_2\ 4\ L\_3;4\ L\_3\ \rm L\_1\tag{21}$$

and those concerning the pressure for nodes 1, 2, 3 are L1, L2, L3, (Figure 6).

The element shown satisfies the LBB condition and thus gives reliable and convergent solutions for velocity and pressure fields [16].

Figure 6. Area coordinates.

Along the centerline of the cylindrical tube (r = 0): vr = 0 ∂vz

∂r ¼ 0.

Vsηo

τrr ¼ �p þ ηð Þ A1 rr þ K 0:85υ1 A2

τθθ ¼ �p þ ηð Þ A1 θθ þ K 0:85υ1 A2

τzz ¼ �p þ ηð Þ A1 zz þ K 0:85υ1 A2

are then back-calculated after finding the velocities vr, vz and the pressure p.

τrz ¼ ηð Þ A1 rz þ K 0:85υ1 A2

3. Application of the finite element method to fluids

p ¼ 0 atmospheric pressure

On the surface of the sphere: vr = vz = 0.

2.4. Dimensionless stress components

η0 Vs

formulas and Eq. (7), τ∗ ¼ τ a

notation by implication as follows:

At the outlet of the flow: vr = 0 ∂vz

Introducing K ¼ υ10

72 Polymer Rheology

primitive variables.

element.

niques are used.

3.1. Linear and quadratic triangles

∂r ¼ 0

a in the CEF constitutive equation Eq. (1), using Eq.(13) nondimensional

rr � 1 2

υ1ð Þ A2 rz

θθ � 1 2

zz � 1 2

1

> 1

1

rz � 1 2

1

Due to the assumptions Eq. (12), only vr, vz velocity components and pressure p are chosen as

All stress variables are expressed in terms of the velocities and the pressures and the stresses

For stability the pressure field must be interpolated with a polynomial one order lower than the velocity terms. Here we have chosen linear pressure and quadratic velocity fields over the

A quadratic triangle for velocities with six nodes, three on the vertices and the three others on mid-sides, and a linear triangle for pressures, both processed with area coordinates (Figure 5). In our problem gravitational and inertial forces are neglected and the motion is obtained as in the first Stokes problem by drawing the cylinder opposite to the sphere displacement with a constant, upward Vs velocity. To solve the nonlinear equations iterative optimization tech-

The Ne f g shape functions for velocities corresponding to the three vertice nodes and three midside nodes (1, 2, …, 6) are respectively using the well-known area coordinates L1, L2, L3.

(19)

stress tensor components can be written electing the \*

υ1ð Þ A2 rr

υ1ð Þ A2 θθ

(20)

υ1ð Þ A2 zz

The derivatives of the interpolation functions with respect to the global coordinates are of the form

$$\begin{Bmatrix} \frac{\partial N\_i^\epsilon}{\partial r} \\\\ \frac{\partial N\_i^\epsilon}{\partial z} \end{Bmatrix} = J^{-1} \begin{Bmatrix} \frac{\partial N\_i^\epsilon}{\partial L\_1} \\\\ \frac{\partial N\_i^\epsilon}{\partial L\_2} \end{Bmatrix} \tag{22}$$

we eliminate the dependent variable L3 for the sake of simplicity. Thus, we have

1 þ 4L1L2 þ 2L2

2 vr5 þ 4 L1 � L1L2 � L2

1 þ 4L1L2 þ 2L2

2 vz5 þ 4 L1 � L1L2 � L2

þ L2 p2 � p3

The flow domain is meshed using linear and quadratic triangular elements [9]. Three unstructured meshes are generated by an adaptive mesh generator, as given in Table 2 and shown in Figure 7. Independent unknown variables are, due to axisymmetry vr, vz and p. The continuity equation being interpreted as an additional relation among the velocity components, i.e., a constraint satisfied in an approximate least squares sense [18], it may be omitted in the effective area consideration around the mid-side nodes, in order to keep the balance between the equation and unknown variable numbers. Resolving the global equation system obtained while taking into account the boundary conditions and integrating numerically by means of Gauss quadrature over the effective area around each node, the values of the variables are

a ¼ 0:05 m a=R ¼ 0:2 Vs ¼ 0:016 m=s

In order to provide a basis for a comparison, and to pose the behavioral difference between inelastic and viscoelastic fluids, an example taken from the literature [9], for the simpler case υ1 = 0, gives contours of streamfunction, velocity, stress, pressure and viscosity at parameter settings of n = 0.5 and We = Weissenberg number = 2.5 (Figure 8). As it can be seen in the

Mesh Sphere surface nodes Elements Nodes AM1 21 644 1401 AM2 31 948 2035 AM3 41 1242 2645

The extrema contour levels for the inelastic fluid are again shown in a table (Table 3).

2 vr2 þ ð1 � 3L1�

2 vz2 þ ð1 � 3L1�

2Þvr3 þ 4L1L2vr4þ

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model

2Þvz3 þ 4L1L2vz4þ

vr6

vz6

1

1

þ p3 (24)

http://dx.doi.org/10.5772/intechopen.75977

75

1 vr1 þ �L2 þ 2L2

þ 4 L2 � L1L2 � L2

1 vz1 þ �L2 þ 2L2

þ 4 L2 � L1L2 � L2

p ¼ L1 p1 � p3

As a typical numerical example after [9], the values below are processed:

Figure 8, the flow accelerates as the fluid approaches the sphere.

vr ¼ �L1 þ 2L2

vz ¼ �L1 þ 2L2

3.3. Basis of application

found [19, 20].

3.3.1. Comparison and contour patterns

Table 2. Summary of finite element meshes.

� 3L2 þ 2L2

� 3L2 þ 2L2

with the Jacobian matrix of transformation

$$J = \begin{vmatrix} \frac{\partial r}{\partial L\_1} & \frac{\partial z}{\partial L\_1} \\ \frac{\partial r}{\partial L\_2} & \frac{\partial z}{\partial L\_2} \end{vmatrix}$$

The element area

$$drdz = \frac{1}{2} \text{det} \mathbf{J} dL\_1 dL\_2 \tag{23}$$

The integration must be carried out over the elemental volume of the axisymmetric geometry rdrdθdz. Since the solution is independent of the θ coordinate, the integration with respect to θ yields a multiplicative constant 2π. (Table 1) [17].

#### 3.2. Interpolation formulas

According to the constraint

$$L\_1 + L\_2 + L\_3 = 1$$

Table 1. Quadrature weights and points for triangular elements.

we eliminate the dependent variable L3 for the sake of simplicity. Thus, we have

$$\begin{aligned} v\_r &= \left(-L\_1 + 2L\_1^2\right)v\_{r1} + \left(-L\_2 + 2L\_2^2\right)v\_{r2} + (1 - 3L\_1 - 2L\_2)v\_{r3} \\ &- 3L\_2 + 2L\_1^2 + 4L\_1L\_2 + 2L\_2^2)v\_{r3} + 4L\_1L\_2v\_{r4} + \\ &+ 4\left(L\_2 - L\_1L\_2 - L\_2^2\right)v\_{r5} + 4\left(L\_1 - L\_1L\_2 - L\_1^2\right)v\_{r6} \\\ v\_z &= \left(-L\_1 + 2L\_1^2\right)v\_{z1} + \left(-L\_2 + 2L\_2^2\right)v\_{z2} + (1 - 3L\_1 - \\ &- 3L\_2 + 2L\_1^2 + 4L\_1L\_2 + 2L\_2^2)v\_{z3} + 4L\_1L\_2v\_{z4} + \\ &+ 4\left(L\_2 - L\_1L\_2 - L\_2^2\right)v\_{z5} + 4\left(L\_1 - L\_1L\_2 - L\_1^2\right)v\_{z6} \\\ p &= L\_1(p\_1 - p\_3) + L\_2\left(p\_2 - p\_3\right) + p\_3 \end{aligned} \tag{24}$$

#### 3.3. Basis of application

The derivatives of the interpolation functions with respect to the global coordinates are of the form

∂Ne i ∂L1 ∂Ne i ∂L2

∂z ∂L1 � � � � � � � � �

detJdL1dL2 (23)

L1 L2 L3 W Geometric locations

1/3 1/3 1/3 1

1/2 0 1/2 0 1/2 1/2 1/2 1/2 0

1/3

∂z ∂L2 9 >>>=

>>>;

(22)

8 >>><

>>>:

∂Ne i ∂r ∂Ne i ∂z

9 >>=

>>; ¼ J �1

∂r ∂L1

� � � � � � � � �

drdz ¼ 1 2

The integration must be carried out over the elemental volume of the axisymmetric geometry rdrdθdz. Since the solution is independent of the θ coordinate, the integration with respect to θ

L1 þ L2 þ L3 ¼ 1

Number of integration points Degree of polynomial Order of the residual Location of integration points

∂r ∂L2

8 >><

>>:

J ¼

with the Jacobian matrix of transformation

yields a multiplicative constant 2π. (Table 1) [17].

1 O (h2 )

2 O (h3 )

Table 1. Quadrature weights and points for triangular elements.

The element area

74 Polymer Rheology

3.2. Interpolation formulas

According to the constraint

1 (Pressures)

3 (Velocities) The flow domain is meshed using linear and quadratic triangular elements [9]. Three unstructured meshes are generated by an adaptive mesh generator, as given in Table 2 and shown in Figure 7. Independent unknown variables are, due to axisymmetry vr, vz and p. The continuity equation being interpreted as an additional relation among the velocity components, i.e., a constraint satisfied in an approximate least squares sense [18], it may be omitted in the effective area consideration around the mid-side nodes, in order to keep the balance between the equation and unknown variable numbers. Resolving the global equation system obtained while taking into account the boundary conditions and integrating numerically by means of Gauss quadrature over the effective area around each node, the values of the variables are found [19, 20].

As a typical numerical example after [9], the values below are processed:

$$a = 0.05 \text{ m} \quad a/R = 0.2 \quad \text{Vs} = 0.016 \text{ m/s}$$

#### 3.3.1. Comparison and contour patterns

In order to provide a basis for a comparison, and to pose the behavioral difference between inelastic and viscoelastic fluids, an example taken from the literature [9], for the simpler case υ1 = 0, gives contours of streamfunction, velocity, stress, pressure and viscosity at parameter settings of n = 0.5 and We = Weissenberg number = 2.5 (Figure 8). As it can be seen in the Figure 8, the flow accelerates as the fluid approaches the sphere.

The extrema contour levels for the inelastic fluid are again shown in a table (Table 3).

Table 2. Summary of finite element meshes.

Figure 7. Mesh patterns around sphere, a/R = 0.2.


**9. Climate change and alters in agro-climatic zoning**

According to the prediction of climate models that predict the continuation of more heat than the global average for high latitudes and elevated areas, especially in winter, as well as the increase in winter temperature and night temperature than in the past, certainly growing domains including natural and hand planting will be affected. Also, the changes that have taken place in agroclimatic indices in the world can be effective. New climatic conditions different from previous ones can alter the niche space (Ecological nest) of any living organism and disturb the favorable biological conditions for living organisms, including plants and animals. This change in environmental conditions, especially changing climatic conditions, may not be desirable for plant species. So that, by changing the optimum climatic conditions of the plant, new plant species to replace the previous species. Chamura et al. [46] used climate models and concluded that agro-climatic zones associated with major food crops of corn, sorghum, cassava, and peanuts in Ghana have changed. Research by Tranka et al. [47] indicates the deterioration of agricultural conditions in Ghana. Their results confirm that dryland farming in this country is facing serious risks and there is an increased risk of very undesirable years in many climatic zones of this country, leading to annual variation of crop yields. Ceglar et al. [48] studied the effect of climate change on the displacement of agro-climatic zones in Europe. The gradual warming of the European continent has led to the prolonged growing season, the cumulative increase in active temperatures combined with the events of the extreme hot events. This research, which has been conducted using climatic models and different emission scenarios, shows that much of Europe will be affected by the displacement of climate regions facing the north in the coming

decades. In addition, the displacement of agro-climatic zones in Eastern Europe may reach twice the speed observed in the period 1975–2016. Some regions may lose the ability to grow specific crops for the benefit of northern European regions. This index-based assessment shows that the potential benefits of prolonging the thermal growing season in northern and eastern Europe are often unbalanced by the risk of late frost and heatwaves in early spring and summer.

Research reveals that due to the shortening of the growing period or due to lack of thermal requirements, agro-climatic zones have been displaced [49, 50–53]. The movement of vineyards from the main planting areas, i.e. the Mediterranean regions to the central and western regions of Europe, is evidence of this claim.

King et al. [54] indicated that suitable conditions for the degree-day of plant growth will experience a north-facing expansion of up to 1200 kilometers in northern regions by the end of the twenty-first century. The northerly expansion will provide favorable conditions for crop production, along with the relocation of agro-climatic zones to inland and other parts of Europe. Although ecological barriers (such as mountains and lakes) can temporarily stop movement, the speed of movement across the boundaries of climatic zones has a significant impact on the redistribution of ecological communities [55].


**4. C3S microarchitecture model**

#### **4.1 Mini-columns and TNNs (completed work)**

We have developed a microarchitecture model for building highly efficient TNN designs [11]. In this model, values are encoded as timings of events or spikes. Spikes are implemented as logic pulses whose timings are calibrated using unit hardware clock. We refer to this as "direct" implementation as hardware clock itself defines the time unit for temporal processing and spikes directly arrive at precisely timed clock cycles. Here, spike timings are not encoded as binary values propagated in the form of packets. The proposed TNN microarchitecture model consists of the following key modules: (1) a counter-based synapse that performs temporal processing of input spikes to generate specialized responses and unsupervised/ supervised STDP learning of synaptic weight; (2) a multi-synapse neuron that accumulates the synaptic responses and fires an output spike when the accumulated response crosses a threshold; and (3) a multi-neuron TNN column (referred henceforth as "mini-column" to distinguish from cortical column) that applies winner-takeall (WTA) inhibition across its neurons. **Figure 6** illustrates the implementation of a mini-column.

A mini-column with *p* synaptic inputs feeding *q* neurons via a *pxq* synaptic crossbar is referred to as a *(pxq) mini-column* in **Figure 6**. The figure illustrates the actual implementation of the major components of a (*pxq*) mini-column, including the synapses, neuron body, STDP local learning, and WTA lateral inhibition. In our prior work [11], we have taken the RTL designs of various configurations of the mini-column through both the synthesis and physical design tools, and obtained PPA (power, performance, area) results by scaling both *p* and *q*. We observed that power and area scale linearly with the total number of synapses (*pxq*), whereas performance limited by the critical path delay scales logarithmically with the number of synaptic inputs (*p*). We also derived a set of characteristic equations from the gate-level designs that provide qualitative estimation of gate-level PPA complexity for arbitrary mini-column designs.

*Cortical Columns Computing Systems: Microarchitecture Model, Functional Building Blocks… DOI: http://dx.doi.org/10.5772/intechopen.110252*

#### **Figure 6.**

*Microarchitecture model for implementing multi-column and multi-layer TNNs [11]. This figure highlights details of a generic TNN column (or mini-column). Each mini-column consists of p inputs feeding a stack of q neurons via a pxq synaptic crossbar. Each cross point in the pxq crossbar stores a weight value that is updated based on spike timing dependent plasticity (STDP) updating rules. Each neuron performs the weighted sum of its inputs. Each mini-column is also supported by winner-take-all (WTA) inhibition across its neuron outputs that selects the winner neuron. Gate-level implementations of these mini-column components are illustrated.*

Multiple mini-columns can be grouped and organized into a hierarchy of layers to form multi-layer temporal neural networks (TNNs), as shown at the bottom of **Figure 6**. Multiple multi-neuron mini-columns are stacked to form a single layer and multiple multi-column layers are cascaded to form a multi-layer TNN. A TNN is typically bookended by input-encode and output-decode layers, and is effectively a feed-forward network. The mini-column is the fundamental building block for TNNs and can learn to distinguish distinct input patterns. Hence, a single mini-column can be viewed as a fully functioning TNN. **Table 1** in Section 6 demonstrates the efficacy of single mini-column designs in performing unsupervised clustering with minimal PPA complexity.

*All mini-columns are specifically trained for their corresponding benchmarks and achieve competitive performance relative to state-of-the-art. Even the largest mini-column design only consumes 39 μ W.*

**Table 1.**

*Design space exploration for UCR time-series clustering: Using TNN7 macros, PPA [12] for seven sample TNN prototype designs (mini-columns) across diverse synapse counts and application benchmarks from [13].*


**4. The choice of filters**

The filter shape and filtering width are the two free parameters in LES. Each affects the LES results greatly. Designing suitable filter type and filtering width is important to get reasonable results. In dynamic Smagrinsky model and similarity model, the effect of the filtering width has been studied by Lund (Lund 1997), De Stefano and Vasilyev (De Stefano and Vasilyev 2002) etc. In order to separate these two effects from one another, the present study is focus on the importance of the filter shape.

Theoretically, the filtering operation should be repeated every time step because the nonlinear term continuously generates high frequency modes that need to be dissipated (Lund 1997). Depending on the type of filter, the SGS model should be adjusted in order to represent the dynamics of the unresolved scales correctly. Consequently the nature of the LES solution strongly depends on the filter shape. But for the traditional LES, especially for the eddy viscosity model, there is no explicit filtering process during the calculation in spite of the reasons mentioned above, i.e., the simulation result is independent of the filter. In the conventional practice, the filter has been only used as a concept (Fröhlich & Rodi 2001). The effect of the filter shape on LES is rarely discussed in the literature.

On the other hand, a suitable LES model is needed to test different filters. As just mentioned, most traditional eddy viscosity models do not have explicit filtering in the solution procedure. In the similarity models and the dynamical Smagorinsky model the filtering width of the test (second) filter plays a key role besides the filter shape. A more appropriate LES model, which can directly validate different filters, is therefore required. From section 3.2 we found that the filtering plays a key role in TNS. The dynamics of the large scales and the energy budget strongly depend on the filter shape. It is a very good model to study the filter effect.

There have been many filters proposed in the literatures that can be categorized into two groups: *smooth filters* and *discrete filters*. At the early stage of development, LES was mostly

Study of Some Key Issues for

or close to it at the same time.

filter as the physical filter

Applying LES to Real Engineering Problems 39

because for most flows transformation to the spectral space is difficult. People are trying to find a filter that is defined in the physical space while has the property of the spectral filter

Actually the filtering operation (2.1)(Leonard 1974) is a linear spatial averaging operation,

1 1 2 *L I IL I IL IL G G GG*

where *I* is the unity operator. The product of *L***G** and the first few terms of the above expansion (4.2) actually defines a suitable new filter (Domaradzki et al 2002) (the product of *L***G** and the full equation (4.2) is equal to *I*, of course). If only first few terms are selected, the new filter is close to the original filter *L***G** and the extra computation cost is small. When more terms are selected, the new filter is closer to unity. It has less effect on the large scales but needs much more computation time. Domaradzki *et. al*. (Domaradzki et al 2002) found that the combination of the first three terms in Eq. 4.2 is the best choice and denoted this

where '^' is the original Tophat filter or Gaussian filter. One thing need to be pointed out is that the results of the Tophat, Gaussian and Physical filters strongly depend on the filter width . However the objective of present work is to highlight the importance of the filter shape as mentioned above. In the following analysis, the filter width is fixed for these

**k**

5 10 15 20 25 30 10-5

A formal inverse of it in a power series expansion can be expressed as

smooth filters, which is equal to 2 times of grid size .

**E(k)**

10-4

Fig. 4.1. Effect of the smooth filters on the *k*-5/3 spectrum.

10-3

10-2

10-1

100

( ) ( ) ( , '; ) ( ') ' *G D f x L f x Gxx f x dx* (4.1)

(4.2)

ˆ ˆ ˆ 3 3 ˆ ˆˆ *u u uu i ii* (4.3)

**k-5/3 Spectral TopHat Gaussian Physical**

performed in spectral space in which the filters were defined as continuous in the whole domain. These filters are referred to as smooth filters. The most commonly used ones are the Tophat (box), the Gaussian, and the sharp spectral (Fourier cutoff) filters. Recently, the application of LES to solve real engineering problems in the complex flow has become realistic and, in fact, popular because of the urgent need from the industry. Finite difference scheme instead of pseudo-spectral method is now widely adopted in the numerical approach due to its flexibility in handling complex geometry and obtaining high order schemes (Lele 1992, Visbal & Gaitonde 2002). The finite difference discretization scheme, together with the limited grid resolution, can be seen as an implicit filtering as mentioned by many researchers (De Stefano and Vasilyev2002; Fröhlich & Rodi 2001; Lund 2003; Vasilyev 1998). However this kind of implicit filtering has some problems because of the interactions among the modified terms in the governing equations, the numerical error, and the order of the filter, etc.(Lund 2003; Vasilyev 1998). In order to avoid some of these problems, researchers tend to employ the explicit filtering to exert direct influence on the simulation result. These filters are usually defined on several adjacent points and, hence, denoted as discrete filters thereafter. There are several advantages using explicit filter: First it is easier to control the truncation and aliasing errors by removing the high wave number modes which is beyond the bandwidth allowed by the mesh. Second it can dump the oscillation at high frequency which comes from the numerical discretization scheme, boundary condition, etc. The amplitude of these oscillations usually is comparable to or even larger than that of the small scales after sufficiently long computation time, which tends to contaminate the final result of the simulation. By using the same explicit filter, it also makes the comparison with experiment or DNS data more direct.

#### **4.1 Smooth filters**

The smooth filters include the spectral filter, the Gaussian filter, the Tophat filter and those are defined continuously in the whole computation domain. The definitions for the first three can be easily found in some books (Pope 2000). Table 4.1 shows these filter functions in physical and spectral space respectively.

Table 4.1. Smooth filters in physical and spectral space / *c k* .

The main problem for the Tophat and Gaussian filters is that they remove too much energy of the large scales (Domaradzki et al 2002; Yang & Domaradzki 2004). The spectral filter is thought as the best among these three for LES because it keeps all the large scales while removes all the small scales. However filters defined in the physical space are much flexible

performed in spectral space in which the filters were defined as continuous in the whole domain. These filters are referred to as smooth filters. The most commonly used ones are the Tophat (box), the Gaussian, and the sharp spectral (Fourier cutoff) filters. Recently, the application of LES to solve real engineering problems in the complex flow has become realistic and, in fact, popular because of the urgent need from the industry. Finite difference scheme instead of pseudo-spectral method is now widely adopted in the numerical approach due to its flexibility in handling complex geometry and obtaining high order schemes (Lele 1992, Visbal & Gaitonde 2002). The finite difference discretization scheme, together with the limited grid resolution, can be seen as an implicit filtering as mentioned by many researchers (De Stefano and Vasilyev2002; Fröhlich & Rodi 2001; Lund 2003; Vasilyev 1998). However this kind of implicit filtering has some problems because of the interactions among the modified terms in the governing equations, the numerical error, and the order of the filter, etc.(Lund 2003; Vasilyev 1998). In order to avoid some of these problems, researchers tend to employ the explicit filtering to exert direct influence on the simulation result. These filters are usually defined on several adjacent points and, hence, denoted as discrete filters thereafter. There are several advantages using explicit filter: First it is easier to control the truncation and aliasing errors by removing the high wave number modes which is beyond the bandwidth allowed by the mesh. Second it can dump the oscillation at high frequency which comes from the numerical discretization scheme, boundary condition, etc. The amplitude of these oscillations usually is comparable to or even larger than that of the small scales after sufficiently long computation time, which tends to contaminate the final result of the simulation. By using the same explicit filter, it also makes the comparison

The smooth filters include the spectral filter, the Gaussian filter, the Tophat filter and those are defined continuously in the whole computation domain. The definitions for the first three can be easily found in some books (Pope 2000). Table 4.1 shows these filter functions in

> *c c*

*k k*

*k k*

*k*

2 2

The main problem for the Tophat and Gaussian filters is that they remove too much energy of the large scales (Domaradzki et al 2002; Yang & Domaradzki 2004). The spectral filter is thought as the best among these three for LES because it keeps all the large scales while removes all the small scales. However filters defined in the physical space are much flexible

*k*

ˆ( ) exp 24 *k G k*

Spectral space Physical space

*k* . sin ( ) *x*

1 , 0.5 ; ( )

0, *x G x*

6 6 ( ) exp *x G x*

*x*

*otherwise*

2 2

2

*G x*

with experiment or DNS data more direct.

physical and spectral space respectively.

Spectral filter 1, ; ˆ( ) 0,

Tophat filter sin 0.5 ˆ( ) 0.5

*G k*

*G k*

Table 4.1. Smooth filters in physical and spectral space / *c*

**4.1 Smooth filters**

Gaussian filter

because for most flows transformation to the spectral space is difficult. People are trying to find a filter that is defined in the physical space while has the property of the spectral filter or close to it at the same time.

Actually the filtering operation (2.1)(Leonard 1974) is a linear spatial averaging operation,

$$
\overline{f}\left(\mathbf{x}\right) = L\_G\left(f\left(\mathbf{x}\right)\right) = \int\_D G\left(\mathbf{x}, \mathbf{x}'; \overline{\Delta}\right) f\left(\mathbf{x}'\right) d\mathbf{x}' \tag{4.1}
$$

A formal inverse of it in a power series expansion can be expressed as

$$L\_{\rm G}^{-1} = \left(I - \left(I - L\_{\rm G}\right)\right)^{-1} = I + \left(I - L\_{\rm G}\right) + \left(I - L\_{\rm G}\right)^2 + \cdots \tag{4.2}$$

where *I* is the unity operator. The product of *L***G** and the first few terms of the above expansion (4.2) actually defines a suitable new filter (Domaradzki et al 2002) (the product of *L***G** and the full equation (4.2) is equal to *I*, of course). If only first few terms are selected, the new filter is close to the original filter *L***G** and the extra computation cost is small. When more terms are selected, the new filter is closer to unity. It has less effect on the large scales but needs much more computation time. Domaradzki *et. al*. (Domaradzki et al 2002) found that the combination of the first three terms in Eq. 4.2 is the best choice and denoted this filter as the physical filter

$$
\overline{\mu} = \Im \hat{u}\_i - \Im \hat{\hat{u}}\_i + \hat{\hat{\hat{u}}}\_i \tag{4.3}
$$

where '^' is the original Tophat filter or Gaussian filter. One thing need to be pointed out is that the results of the Tophat, Gaussian and Physical filters strongly depend on the filter width . However the objective of present work is to highlight the importance of the filter shape as mentioned above. In the following analysis, the filter width is fixed for these smooth filters, which is equal to 2 times of grid size .

Fig. 4.1. Effect of the smooth filters on the *k*-5/3 spectrum.

Study of Some Key Issues for

**E(k)**

downwards periodically.

**K(t)**

0.2

Fig. 4.4. The decay of the total kinetic energy for the smooth filters.

0.3

0.4

0.5

0.6

0.7

10-3

Fig. 4.3. Energy spectrum at final time for LES case 2.

10-2

10-1

Applying LES to Real Engineering Problems 41

**k**

**DNS Spectral Tophat Physical**

5 10 15 20 25 30 10-4

The history of total kinetic energy decay for the smooth filters is plotted in Fig. 4.4. Corresponding to Fig. 4.3, the physical filter obtains better result compared to DNS. Since the spectral filter does not provide sufficient dissipation as shown in Fig. 4.3, its total energy is the biggest among all the results. The Tophat filter removes a part of large-scale energy each time when the filtering operation is applied. That is why the total energy jumps

**t**

0 0.511.5 0.1

**Initial Condition**

**DNS Spectral Tophat Physical**

The effects of above filters on the *k***-5/3** spectrum are shown in Fig.4.1. As can be seen, the Tophat and Gaussian filters remove too much energy of the low modes. The spectral filter only keeps the large scales. The physical filter strongly damps the small scales while affecting the large scales very little, which make it a good filter for LES.

Beside *a prior* test, the effects of these filters on a real three dimensional LES are also examined. Again the simplest homogeneous, isotropic decaying turbulence is utilized as the test case with two different initial conditions. The first has the initial condition of spectrum 4 22 *E k Ak k k* ( ) exp 2 / *p* , where *k***p** is the peak mode and equals to 4. The grid resolution is 643 (In the following, the mesh size is 643 for all LES unless further specified). For comparison, the 2563 DNS result is also included. The final energy spectrum is plotted in Fig. 4.2. It also shows that the Tophat filter removes too much energy (Since the Gaussian filter performs very similarly to the Tophat filter, we did not include it in the figure). The spectral and physical filters show very good agreement with the DNS data.

Fig. 4.2. Energy spectrum at final time for LES case 1.

The second case has a more critical initial condition as shown in Fig. 4.3. The initial condition is obtained from the 2563 DNS data of Horiuti (Horiuti 1999) same as section 3. It is a challenging case for LES because the energy at cut-off mode *k*c is not in the inertial range. The Tophat filter shows too much dissipation same as above. However the spectral filter delivers some undesirable behaviors this time. By removing all the small scales, it also shut down the energy transfer from the large to small scales completely. It will take some time for LES to rebuild the nonlinear interactions between the large and small scales, which leads to insufficient dissipation. Thus, the energy accumulates near the cutoff wavenumber as shown in Fig. 4.3. The physical filter provides the best result compared to the DNS data as also observed in paper ((Domaradzki et al 2002; Yang & Domaradzki 2004). The main reason is that the physical filter keeps a small part of the small scales which facilitates the energy transfer. Since this initial condition is a better case to test filters, we only run LES with case two in the following discussion.

The effects of above filters on the *k***-5/3** spectrum are shown in Fig.4.1. As can be seen, the Tophat and Gaussian filters remove too much energy of the low modes. The spectral filter only keeps the large scales. The physical filter strongly damps the small scales while

Beside *a prior* test, the effects of these filters on a real three dimensional LES are also examined. Again the simplest homogeneous, isotropic decaying turbulence is utilized as the test case with two different initial conditions. The first has the initial condition of spectrum 4 22 *E k Ak k k* ( ) exp 2 / *p* , where *k***p** is the peak mode and equals to 4. The grid resolution is 643 (In the following, the mesh size is 643 for all LES unless further specified). For comparison, the 2563 DNS result is also included. The final energy spectrum is plotted in Fig. 4.2. It also shows that the Tophat filter removes too much energy (Since the Gaussian filter performs very similarly to the Tophat filter, we did not include it in the figure). The

**k**

5 10 15 20 25 30 10-4

The second case has a more critical initial condition as shown in Fig. 4.3. The initial condition is obtained from the 2563 DNS data of Horiuti (Horiuti 1999) same as section 3. It is a challenging case for LES because the energy at cut-off mode *k*c is not in the inertial range. The Tophat filter shows too much dissipation same as above. However the spectral filter delivers some undesirable behaviors this time. By removing all the small scales, it also shut down the energy transfer from the large to small scales completely. It will take some time for LES to rebuild the nonlinear interactions between the large and small scales, which leads to insufficient dissipation. Thus, the energy accumulates near the cutoff wavenumber as shown in Fig. 4.3. The physical filter provides the best result compared to the DNS data as also observed in paper ((Domaradzki et al 2002; Yang & Domaradzki 2004). The main reason is that the physical filter keeps a small part of the small scales which facilitates the energy transfer. Since this initial condition is a better case to test filters, we only run LES

**Initial Condition**

**DNS Spectral Tophat Physical**

affecting the large scales very little, which make it a good filter for LES.

spectral and physical filters show very good agreement with the DNS data.

**E(k)**

10-3

with case two in the following discussion.

Fig. 4.2. Energy spectrum at final time for LES case 1.

10-2

10-1

Fig. 4.3. Energy spectrum at final time for LES case 2.

The history of total kinetic energy decay for the smooth filters is plotted in Fig. 4.4. Corresponding to Fig. 4.3, the physical filter obtains better result compared to DNS. Since the spectral filter does not provide sufficient dissipation as shown in Fig. 4.3, its total energy is the biggest among all the results. The Tophat filter removes a part of large-scale energy each time when the filtering operation is applied. That is why the total energy jumps downwards periodically.

Fig. 4.4. The decay of the total kinetic energy for the smooth filters.

Study of Some Key Issues for

utilized rather than the spectral smooth filters

1 1/4 1/2 1/4

Table 4.2. The weight parameters of the V-filters.

3 1/8 5/8 3/8 -1/8 6 -1/16 1/4 5/8 1/4 -1/16

9 -1/32 5/32 11/16 5/16 -5/32 1/32 10 1/64 -3/32 15/64 11/16 15/64 -3/32 1/64

order are 3 and 5 respectively) are tested as well as the symmetric ones.

**4.2 Discrete filters**

(Mathew *et al.* 2003).

Applying LES to Real Engineering Problems 43

In order to handle complex geometry, finite difference scheme is widely used instead of the spectral method. The solution is available only on a set of discrete grid points. At most time, the filter for the whole domain does not exist due to the inhomogeneous and boundary condition. The discrete filters, including the discrete Tophat filter, the Padé filter (Visbal & Gaitonde 2002) and the filter series proposed by Vasilyev *et al.* (Vasilyev *et al.* 1998) are

The main problem of the discrete filter is the commutation error between differentiation and filtering operation. Fortunately Vasilyev *et al.* (Vasilyev *et al.* 1998) gave out a solution which can control the commutation error to any specified order. Another problem is that if the order of the filter is too low, the error introduced by filtering may become larger than the magnitude of SGS term. Hence for traditional LES, the filter order is usually required to be higher than that of SGS term. But the filtering operation in TNS only acts as a dissipation source. The numerical error can be included into it as part of the dissipation. Low order filters can also obtain good results. It is similar to the strategy used by Mathew *et al.*

A one-dimensional filter given by Vasilyev *et al.* (Vasilyev *et al.* 1998) is defined as:

*L j l j l l K f w f*

In order to control the commutation error to a specified order, the filter is required to have a different number of vanishing moments. Correspondingly, the weight factors *w***i** should satisfy a set of constrains. These filters are referred as V-filters in the following analysis.

Case *w*3 *w*2 *w*1 *w*0 *w*1 *w*2 *w*3 *w*4 *w*5

7 31/32 5/32 -5/16 5/16 -5/32 1/32

Several sets of weights for the V-filters are given in Table 4.2 which is similar to the Table 1 in Vasilyev *et al*'s paper (Vasilyev *et al.* 1998). The equation (4.4) defines a symmetric (center) scheme if K equals to L. The case 1, 6 and 10 are symmetric and have a commutation error of order 2, 4, 6 respectively. In order to handle boundary points, Vasilyev *et al* also proposed several asymmetric filters, i.e., K and L is different. For high asymmetric V-filters, such as the one side filter - case 7, it is found that too much unphysical energy is introduced to the high modes as also mentioned by Vasilyev *et al* (Vasilyev *et al.* 1998). This property is not desirable for TNS because it will lead to unphysical solution. Thus high asymmetric filters (case 2, 4 5, 7, 8) are not included in the following analysis. Only case 3 and 9 (whose

Fig. 4.7 presents the filtering results of different V-filters applied to the *k***-5/3** spectrum. For comparison, the result of the smooth physical filter is also included. Case 1 in fact is a discrete version of the Tophat filter using trapezoidal rule. Similar to the smooth one, it

(4.4)

The effect of different grid resolutions is also investigated. The result of LES with grid 323 is shown in Fig4.5 and the result for 1283 is plotted in Fig. 4.6. The behaviors of these smooth filters in coarse mesh (323) are almost the same as those in grid 643. Physical filter still gets the best results. For the fine mesh (1283), all filters obtain good result except that the Tophat filter still dissipates a little more. The effect of the SGS model becomes small when the grid resolution increases, which is well known.

Fig. 4.5. Energy spectrum at final time for the smooth filters with grid 323.

Fig. 4.6. Energy spectrum at final time for the smooth filters with grid 1283.

### **4.2 Discrete filters**

42 Computational Simulations and Applications

The effect of different grid resolutions is also investigated. The result of LES with grid 323 is shown in Fig4.5 and the result for 1283 is plotted in Fig. 4.6. The behaviors of these smooth filters in coarse mesh (323) are almost the same as those in grid 643. Physical filter still gets the best results. For the fine mesh (1283), all filters obtain good result except that the Tophat filter still dissipates a little more. The effect of the SGS model becomes small when the grid

**k**

**k**

10 20 30 40 5060 10-4

**Initial Spectrum**

**DNS Spectral Tophat Physical**

Fig. 4.6. Energy spectrum at final time for the smooth filters with grid 1283.

5 10 15 10-4

**Initial Spectrum**

**DNS Spectral Tophat Physical**

Fig. 4.5. Energy spectrum at final time for the smooth filters with grid 323.

resolution increases, which is well known.

10-1

**E(k)**

10-3

**E(k)**

10-3

10-2

10-1

10-2

In order to handle complex geometry, finite difference scheme is widely used instead of the spectral method. The solution is available only on a set of discrete grid points. At most time, the filter for the whole domain does not exist due to the inhomogeneous and boundary condition. The discrete filters, including the discrete Tophat filter, the Padé filter (Visbal & Gaitonde 2002) and the filter series proposed by Vasilyev *et al.* (Vasilyev *et al.* 1998) are utilized rather than the spectral smooth filters

The main problem of the discrete filter is the commutation error between differentiation and filtering operation. Fortunately Vasilyev *et al.* (Vasilyev *et al.* 1998) gave out a solution which can control the commutation error to any specified order. Another problem is that if the order of the filter is too low, the error introduced by filtering may become larger than the magnitude of SGS term. Hence for traditional LES, the filter order is usually required to be higher than that of SGS term. But the filtering operation in TNS only acts as a dissipation source. The numerical error can be included into it as part of the dissipation. Low order filters can also obtain good results. It is similar to the strategy used by Mathew *et al.* (Mathew *et al.* 2003).

A one-dimensional filter given by Vasilyev *et al.* (Vasilyev *et al.* 1998) is defined as:

$$\overline{f}\_{j} = \sum\_{l=-K}^{L} w\_{l} f\_{j+l} \tag{4.4}$$

In order to control the commutation error to a specified order, the filter is required to have a different number of vanishing moments. Correspondingly, the weight factors *w***i** should satisfy a set of constrains. These filters are referred as V-filters in the following analysis.

Table 4.2. The weight parameters of the V-filters.

Several sets of weights for the V-filters are given in Table 4.2 which is similar to the Table 1 in Vasilyev *et al*'s paper (Vasilyev *et al.* 1998). The equation (4.4) defines a symmetric (center) scheme if K equals to L. The case 1, 6 and 10 are symmetric and have a commutation error of order 2, 4, 6 respectively. In order to handle boundary points, Vasilyev *et al* also proposed several asymmetric filters, i.e., K and L is different. For high asymmetric V-filters, such as the one side filter - case 7, it is found that too much unphysical energy is introduced to the high modes as also mentioned by Vasilyev *et al* (Vasilyev *et al.* 1998). This property is not desirable for TNS because it will lead to unphysical solution. Thus high asymmetric filters (case 2, 4 5, 7, 8) are not included in the following analysis. Only case 3 and 9 (whose order are 3 and 5 respectively) are tested as well as the symmetric ones.

Fig. 4.7 presents the filtering results of different V-filters applied to the *k***-5/3** spectrum. For comparison, the result of the smooth physical filter is also included. Case 1 in fact is a discrete version of the Tophat filter using trapezoidal rule. Similar to the smooth one, it

Study of Some Key Issues for

**K(t)**

0.2

Fig. 4.9. The decay of the total kinetic energy for the V-filters.

For a variable *f*, the filtered value can be expressed as:

coefficients *na* are listed in Table 4.3.

where  0.3

0.4

0.5

0.6

0.7

Applying LES to Real Engineering Problems 45

Fig.4.8 shows the final energy spectrum of the V-filters in the same homogeneous run as Fig.4.3. As expected, case 1 dissipates too much energy. High order filters obtains better results. Corresponding to Fig.4.7, the result of the low order asymmetric filter case 3 (3rd order) is a little better than that of the high order symmetric filter case 6 (4th order). And the behaviors of case9 and case 10 are very similar. The reason is still attributed to the fact that a small amount of energy is introduced at high modes for the asymmetric filters. The decay of the total energy is plotted in Fig.4.9. Except case 1, all other cases show good agreement with the filtered DNS data. But there are small jumps for low order filters (case 3 and case 6) because of the undesirable effect to the large scales as shown in Fig.4.7. The effects of grid

resolutions on V-filters are similar to the smooth ones, which are not included here.

**t**

1 1 0 2

*a*

(4.5)

*f* means a less

*N n f i i f i i n i n n*

 *ff f ff*

dissipative filter. N is the order of filter scheme, 2N+1 points give a 2N order filter. The

The filtering results of the *k*-5/3 spectrum using different Padé filters are shown in Fig. 4.10. The smooth physical filter is also included as a benchmark. The 2nd order filter removes a small amount of the energy of the low modes, which may have undesirable effect on LES because of the unnecessary dissipation. The 4th order and above have little effect on the

*f* is an adjustable parameters between (-0.5, 0.5) and high value of

**DNS case1 case3 case6 case9 case10**

0 0.511.5 0.1

Another series of discrete filters is the Padé filters. The Padé compact difference scheme can be regarded as an implicit filter (Visbal & Gaitonde 2002; Vasilyev et al 1998). Based on that, a set of Padé explicit filters is proposed by Visbal and Gaitonde (Visbal & Gaitonde 2002).

removes too much energy of the large scales. It is interesting that the asymmetric filters like case 3 and 9 keep more energy than the symmetric ones (case 6 and 10). Note case 3 and case 6 still remove a small part of the large-scale energy. While case 7 introduces too much energy at high wavenumber modes which will lead to an unphysical solution in a real LES run.

Fig. 4.7. Effect of the V-filters on the *k*-5/3 spectrum.

Fig. 4.8. Energy spectrum at final time for the V-filters.

removes too much energy of the large scales. It is interesting that the asymmetric filters like case 3 and 9 keep more energy than the symmetric ones (case 6 and 10). Note case 3 and case 6 still remove a small part of the large-scale energy. While case 7 introduces too much energy at high wavenumber modes which will lead to an unphysical solution in a real LES

**k**

**k**

5 10 15 20 25 30 10-4

**Initial Spectrum**

**DNS case 1 case 3 case 6 case 9 case 10**

5 10 15 20 25 30

**k-5/3 Physical case 1 case 3 case 6 case 7 case 9 case 10**

run.

**E(k)**

10-4

**E(k)**

10-3

Fig. 4.8. Energy spectrum at final time for the V-filters.

10-2

10-1

Fig. 4.7. Effect of the V-filters on the *k*-5/3 spectrum.

10-3

10-2

10-1

100

Fig.4.8 shows the final energy spectrum of the V-filters in the same homogeneous run as Fig.4.3. As expected, case 1 dissipates too much energy. High order filters obtains better results. Corresponding to Fig.4.7, the result of the low order asymmetric filter case 3 (3rd order) is a little better than that of the high order symmetric filter case 6 (4th order). And the behaviors of case9 and case 10 are very similar. The reason is still attributed to the fact that a small amount of energy is introduced at high modes for the asymmetric filters. The decay of the total energy is plotted in Fig.4.9. Except case 1, all other cases show good agreement with the filtered DNS data. But there are small jumps for low order filters (case 3 and case 6) because of the undesirable effect to the large scales as shown in Fig.4.7. The effects of grid resolutions on V-filters are similar to the smooth ones, which are not included here.

Fig. 4.9. The decay of the total kinetic energy for the V-filters.

Another series of discrete filters is the Padé filters. The Padé compact difference scheme can be regarded as an implicit filter (Visbal & Gaitonde 2002; Vasilyev et al 1998). Based on that, a set of Padé explicit filters is proposed by Visbal and Gaitonde (Visbal & Gaitonde 2002). For a variable *f*, the filtered value can be expressed as:

$$a\sigma\_f \overline{f}\_{i-1} + \overline{f}\_i + a\sigma\_f \overline{f}\_{i+1} = \sum\_{n=0}^{N} \frac{a\_n}{2} (f\_{i+n} + f\_{i-n}) \tag{4.5}$$

where *f* is an adjustable parameters between (-0.5, 0.5) and high value of *f* means a less dissipative filter. N is the order of filter scheme, 2N+1 points give a 2N order filter. The coefficients *na* are listed in Table 4.3.

The filtering results of the *k*-5/3 spectrum using different Padé filters are shown in Fig. 4.10. The smooth physical filter is also included as a benchmark. The 2nd order filter removes a small amount of the energy of the low modes, which may have undesirable effect on LES because of the unnecessary dissipation. The 4th order and above have little effect on the

Study of Some Key Issues for

**E(k)**

**K(t)**

0.2

Fig. 4.12. The decay of the total kinetic energy for the Padé filters.

0.3

0.4

0.5

0.6

0.7

10-3

Fig. 4.11. Energy spectrum at final time for the Padé filters.

10-2

10-1

PV-filter.

Applying LES to Real Engineering Problems 47

infeasible for the inhomogeneous case. In the Section 4.1 the physical filter shows very good property but it is a smooth filter. So we modified it into a discrete version using the V-filters, i.e. in equation 4.3 we use the V-filters instead of the Tophat filter. Hereafter we denote it as

**k**

**DNS Pade 2nd Pade 4th Pade 6th Pade 8th Pade 10th**

**t**

0 0.511.5 0.1

5 10 15 20 25 30 10-4

**Initial Spectrum**

**DNS Pade2 Pade4 Pade6 pade8 Pade10**

Fig. 4.10. Effect of the V-filters on the *k*-5/3 spectrum.

Table 4.3. The Coefficients of the Padé filters.

large scales. But higher the order is, the Padé filter tends to keep more small scales compared to the physical filter. In turn it may not provide enough dissipation for TNS.

The final energy spectra of the LES run with the Padé filters are plotted in Fig. 4.11 and the time evolutions of the total energy are shown in Fig. 4.12. Corresponding to Fig.4.10 the 2nd order filter overestimates the dissipation and subsequently provides the worst results among these runs. The 4th order and above show very good results as compared to the DNS data. However the 6th order and above filters keep more small-scale energy than DNS which may imply them do not provide enough dissipation.

From above results, it is found that the Padé filters show better results than the V-filters. It could be attributed to the fact that the Padé filters consider the effects of adjacent points. On the other hand, the calculation of the V-filters is much simple and straightforward. For the Padé filters we need to solve a tri-diagonal system. It is time consuming and may be

**k-5/3 Physical Pade 2nd Pade 4th Pade 6th Pade 8th Pade 10th**

**E(k)**

10-3

Fig. 4.10. Effect of the V-filters on the *k*-5/3 spectrum.

1

1 2 *f a*

15 17 32 16 *f a*

16

256

which may imply them do not provide enough dissipation.

*f a* 7 18

*f a* 105 302

Table 4.3. The Coefficients of the Padé filters.

F2 1

F4 5 3

F6 11 5

F8

F10

2 *f a*

8 4 *f a*

16 8 *f a*

93 70 128

193 126 256

10-2

10-1

100

**k**

5 10 15 20 25 30 10-4

Scheme 0*a* 1*a* 2 *a* 3 *a* 4 *a* 5 *a*

1 1

3 3 16 8 *f a*

32

64

large scales. But higher the order is, the Padé filter tends to keep more small scales compared to the physical filter. In turn it may not provide enough dissipation for TNS. The final energy spectra of the LES run with the Padé filters are plotted in Fig. 4.11 and the time evolutions of the total energy are shown in Fig. 4.12. Corresponding to Fig.4.10 the 2nd order filter overestimates the dissipation and subsequently provides the worst results among these runs. The 4th order and above show very good results as compared to the DNS data. However the 6th order and above filters keep more small-scale energy than DNS

From above results, it is found that the Padé filters show better results than the V-filters. It could be attributed to the fact that the Padé filters consider the effects of adjacent points. On the other hand, the calculation of the V-filters is much simple and straightforward. For the Padé filters we need to solve a tri-diagonal system. It is time consuming and may be

8 4 *f a*

*f a* 1

*f a* 45(1 2 )

1 32 16

16 8

512

*f a*

*f a* 1

*f a* 5( 1 2 )

128 64 *f a*

256

*f a* 1 2

512 *f a*

2 *f a*

*f a* 7 14

*f a* 15( 1 2 )

infeasible for the inhomogeneous case. In the Section 4.1 the physical filter shows very good property but it is a smooth filter. So we modified it into a discrete version using the V-filters, i.e. in equation 4.3 we use the V-filters instead of the Tophat filter. Hereafter we denote it as PV-filter.

Fig. 4.11. Energy spectrum at final time for the Padé filters.

Fig. 4.12. The decay of the total kinetic energy for the Padé filters.

Study of Some Key Issues for

**E(k)**

**K(t)**

0.2

0.3

0.4

0.5

0.6

0.7

10-3

Fig. 4.14. Energy spectrum at final time for the PV-filters.

10-2

10-1

Applying LES to Real Engineering Problems 49

**k**

**t**

0 0.5 1 1.5 0.1

For DNS, the numerical errors mainly are aliasing and truncation errors (Chow 2003). As for LES, the small scales must be modelled because of the limited grid resolution which can not

**DNS PV-case1 PV-case3 PV-case6 PV-case9 PV-case10**

Fig. 4.15. The decay of the total kinetic energy for the PV-filters.

**5. The effects of numerical errors on TNS**

5 10 15 20 25 30 10-4

**Initial Spectrum**

**DNS PV-case1 PV-case6 PV-case9 PV-case10**

The results of a priori test and the LES run of the PV-filters are shown in Fig. 4.13, Fig. 4.14 and Fig. 4.15 respectively. For comparison, we also include the result of the original case 1 of the V-filter (V-case1) in Fig.4.13. It shows that the result of the 2nd order PVcase1 is improved significantly as compared to V-case1. As mentioned before V-case1 is actually a discrete version of the Tophat filter. Therefore the PV-case1 is a discrete version of the smooth physical filter. Since the result of smooth physical filter is much better than that of the Tophat filter as shown in Fig.4.1, it is no wonder that the PV-case1 is better than the V-case1. However, by comparing Fig.4.14 and Fig.4.3, it was found the results of this discrete version (PV-case1) are not as good as those of the smooth physical filter for a real LES run. Similar to Fig.4.7, the asymmetric filters (PV-case3 and PV-case9) show good behavior in Fig.4.13, keeping more energy than the symmetric ones (PV-case6 and PVcase10). But it was also found that small amount of nonphysical energy is introduced near the cutoff wavenumber. Correspondingly the behaviors of these asymmetric filters in a real LES run are not good as shown in Fig.4.14 and Fig.4.15 (PV-case3 is not shown in Fig.4.14 because the energy spectrum becomes so large that it is out of the scope range). The results of high order symmetric PV-filters are also improved compared to the original V-filters, but not too much.

Fig. 4.13. Effect of the PV-filters on the *k*-5/3 spectrum.

The results of a priori test and the LES run of the PV-filters are shown in Fig. 4.13, Fig. 4.14 and Fig. 4.15 respectively. For comparison, we also include the result of the original case 1 of the V-filter (V-case1) in Fig.4.13. It shows that the result of the 2nd order PVcase1 is improved significantly as compared to V-case1. As mentioned before V-case1 is actually a discrete version of the Tophat filter. Therefore the PV-case1 is a discrete version of the smooth physical filter. Since the result of smooth physical filter is much better than that of the Tophat filter as shown in Fig.4.1, it is no wonder that the PV-case1 is better than the V-case1. However, by comparing Fig.4.14 and Fig.4.3, it was found the results of this discrete version (PV-case1) are not as good as those of the smooth physical filter for a real LES run. Similar to Fig.4.7, the asymmetric filters (PV-case3 and PV-case9) show good behavior in Fig.4.13, keeping more energy than the symmetric ones (PV-case6 and PVcase10). But it was also found that small amount of nonphysical energy is introduced near the cutoff wavenumber. Correspondingly the behaviors of these asymmetric filters in a real LES run are not good as shown in Fig.4.14 and Fig.4.15 (PV-case3 is not shown in Fig.4.14 because the energy spectrum becomes so large that it is out of the scope range). The results of high order symmetric PV-filters are also improved compared to the original

**k**

5 10 15 20 25 30

**k-5/3 Physical PV-case1 PV-case3 PV-case6 PV-case9 PV-case10 original case1**

V-filters, but not too much.

**E(k)**

10-4

Fig. 4.13. Effect of the PV-filters on the *k*-5/3 spectrum.

10-3

10-2

10-1

Fig. 4.14. Energy spectrum at final time for the PV-filters.

Fig. 4.15. The decay of the total kinetic energy for the PV-filters.