**5.2 Total hip arthroplasty injection technique**

A standard approach can be undertaken to local anaesthesia administration intraoperatively can be adopted regardless of approach used (anterior/posterior/ lateral). The primary purpose is to provide anaesthesia to all area affected by the trauma of the surgery. The total volume used intraoperatively of anaesthetic mixture is 150-200 mL where three main stages of administration are involved.

This epidural catheter is then attached to either an antibacterial filter if the intention is post-operative bolus application of anaesthetic mixture or continuous elastomeric pain pump.

### **5.3 Total knee arthroplasty injection technique**

As with hip arthroplasty, in knee arthroplasty the aim is to infiltrate local anaesthetic mixture into areas affected by the surgery. General principals can be broken into four main stages.

away from the popliteal artery. Once the definitive implants are inserted structures around medial collateral ligaments, quadriceps, and tibial portion of the iliotibial band. The epidural catheter is placed through the skin, fascia, quadriceps, and joint capsule using a Tuohy needle, then inserting the tip of the wound catheter posterior to the medial femoral condyle. Also, as was mentioned with hip arthroplasty, care must be taken when closing the wound not to catch catheter with sutures which will impede catheter removal.

An alternate catheter placement can be within the adductor canal, performed intraoperatively. Adductor canal catheter placement may be performed under direct visualisation on subvastus approach. In recovery, ice pack should be applied hourly for 20-minute intervals for the first four hours to prevent swelling and assist in pain control.

#### **5.4 Post-operative**

#### *5.4.1 Continuous infusion vs. bolus in LIA*

The evidence favouring continuous, or bolus post-operative catheter regimes is limited. Early development by Kerr & Kohen used a bolus protocol in THA and TKA, along with other published protocols [10, 14, 42, 49]. Ballarat experience in their study published in ANZ Journal of Surgery in 2020 supported that bolus injection demonstrated better outcome. However, there is also evidence to support the use of continuous infusions of anaesthetic mixture delivered by elastomeric pump devices post operatively which has also shown success [50–52]. Typically, the infusion rate of 5 ml per hour for 48 hours if continuous infusion is used.

Our preference is 25 ml LIA cocktail without adrenaline is injected in the morning after surgery as a bolus.

#### *5.4.2 Post-operative bandage*

Compressive bandage applied post-operatively prolongs the local anaesthetic within the peri-articular tissues after TKA. An elastic binder compressing a sponge on the hip wound could be used but is not common practice and there is no evidence for efficacy. The bandage should be placed to have enough venous compression but not arterial compression [10].

A compression bandage is applied post operatively over the whole leg, from toes to high thigh, in three layers; one layer of soft padding, one layer of crepe bandage, and outer layer of elastic adhesive bandage as recommended by Kerr and Kohan in 2008 [10]. Another study confirmed the benefits of the compression bandage in TKA and demonstrated the improvement of LIA with compression bandage [53]. The compression dressing is removed 24–48 hours post operatively.

#### **6. Pillars of fast track Arthroplasty**

In recent times, many health care systems globally have undergone revision of their hip and knee arthroplasty pathways [40, 54]. The concept of ERAS, 'rapid recovery' or

#### *Fast Track Arthroplasty Using Local Infiltration Analgesia DOI: http://dx.doi.org/10.5772/intechopen.99433*

'fast-tracking' aims to reduce operation-related physiological and psychological stress as well as enhance early mobilisation and reduce recovery time [44]. Implementation of this rapid recovery model has resulted in reduced length of stay with no increase in hospital re-admissions [40, 43]. In recent times, many health care systems globally have undergone revision of their hip and knee arthroplasty pathways [40, 54]. The concept of ERAS, 'rapid recovery' or 'fast-tracking' aims to reduce operation-related physiological and psychological stress as well as enhance early mobilisation and reduce recovery time [44]. Implementation of this rapid recovery model has resulted in reduced length of stay with no increase in hospital re-admissions [40, 43].

Three stages are identified in a patient's journey: pre-operative, peri-operative and post-operative. Pre-operative patient education is a recognised cornerstone in rapid recovery programs [54], with particular attention paid to patient expectation management regarding post-operative pain and LOS. Simultaneously, pre-operative anaesthetic review allows assessment and modification of patient risk factors that have been shown to reduce length of stay in hip and knee arthroplasty patients [55].

#### **6.1 Preoperative**

Rapid recovery protocol begins at the pre-admission stage. On presentation to pre-admission clinic, patients require education on appropriate expectations regarding their operation and post-operative course. An assessment tool in assisting prediction on length of stay and discharge destination (i.e. home or rehabilitation facility) is the Risk Assessment and Prediction Tool (RAPT) which has been previously validated [56, 57]. This tool is a score from 0 to 12, the higher the score the better, where points are given based on age, gender, average walking distance, requirement for walking aids, community, and home supports. With RAPT score < 6 the patient will likely require rehabilitation, between 6 and 9 the patient will likely be discharged home and with a score greater than 9 the patient will likely be discharged post-operative day 1 [57].

There is minimal evidence relating to the impact of preoperative patient education where a Cochrane review by McDonald et al. found preoperative education had no measurable impact on preoperative anxiety or surgical outcomes including pain, function and adverse events [58]. McDonald et al. however noted that there was much heterogeneity and low-level evidence within included studies and preoperative education may benefit patients with depression, unrealistic expectations, anxiety or those with limited social support [58]. The authors believe that setting expectations preoperatively is extremely important for patients prior to their day of surgery where they understand arthroplasty will involve pain which cannot be completely removed, they will be mobilised day of operation, and the primary goal will be discharged home once they are deemed safe. Patients should be prepared to go home post-operative day one.

Pre-admission optimisation is another essential factor when preparing for hip and knee arthroplasty. Assessment and optimisation of risk factors including smoking, alcohol consumption, anaemia, nutritional and metabolic status, and low physical activity has been shown to have a positive impact on length of stay and postoperative complications [59]. Smoking has been shown to increase early post-operative complications, however there is level 2 evidence showing that cessation 4 weeks or greater preoperatively can improve post-operative complications especially related to wound healing [8]. Preoperative optimisation within fast-track protocols has also been shown to reduce the number of patients with a delayed recovery [59].

#### **6.2 Perioperative**

There are several factors that are considered in fast-track arthroplasty regarding preparation for surgery including fasting duration and carbohydrate loading.

#### *Topics in Regional Anesthesia*

Although now relatively common practice, anaesthetic guidelines no longer recommend prolonged fasting status, rather a 2 hour clear and 6 hour solid fasting status prior to surgery [60]. Although a component of fast-track/ERAS protocol, carbohydrate has mixed evidence on impact of clinical outcomes in hip and knee arthroplasty even though there some studies have shown positive impact on postoperative hunger, pain, glucose metabolism and insulin resistance [8].

Standardised anaesthesia is another component included in ERAS protocols with mixed evidence regarding superiority between neuraxial anaesthesia/regional and general anaesthesia [7, 8, 11]. Several studies showed the benefit of neuraxial/ regional anaesthesia including reduced pulmonary compromise, pulmonary embolism, need for transfusion, renal injury, infection, length of stay and 30-day mortality [61, 62]. Alternatively, a systematic review and meta-analysis by Johnston et al. found no difference between neuraxial and general anaesthesia [63]. In general, the aim with fast-track arthroplasty is to reduce systemic opioid use. This avoids associated side effects of opioid analgesia which can inhibit engagement in post-operative recovery by adopting effective multimodal analgesia and anaesthesia.

#### *6.2.1 Tranexamic acid*

There is potential for large blood loss with hip and knee arthroplasty which in turn can prolong hospital stay, increase risk of transfusion, renal failure and increase risk of deep infection. The inclusion of tranexamic acid in hip and knee arthroplasty is effective and safe medication which reduces blood loss with no increased risk of thromboembolic events [7, 8].

#### *6.2.2 Post-operative nausea and vomiting*

Post-operative nausea and vomiting (PONV) can be extremely distressing for patients which consequently impact their post-operative course. Risk factors include narcotics, inhalational anaesthesia, female sex, non-smoking status, history of motion sickness or previous PONV and predicted requirement for post-operative opioids [7]. Active screening and prophylactic treatment for at risk individuals is recommended, but our experience with Total Intravenous Anaesthesia (TIVA) or at least minimising inhalational anaesthesia suggests nausea and vomiting to be less common than it once was.

#### *6.2.3 Active intraoperative warming*

Maintaining normothermia must be a component of anaesthetic care for joint arthroplasty. When normothermia is maintained, it reduces infection, cardiac complications, transfusion requirements and coagulopathy [7]. With joint arthroplasty, aggressive warming has been shown to reduce intraoperative blood loss in total hip arthroplasty and reduce opioid requirements along with improved patient satisfaction in total knee arthroplasty [7]. Several techniques have been described to assist in maintain normothermia including warm IV fluids and irrigation fluid, prewarming and humidification of anaesthetic gases along with forced air-warming blankets and devices [8].

#### *6.2.4 Avoid urinary catheters*

With the use of low dose or no opioid spinal anaesthetic, urinary catheters are unnecessary for the majority of patients, avoiding bacteraemia which may increase

#### *Fast Track Arthroplasty Using Local Infiltration Analgesia DOI: http://dx.doi.org/10.5772/intechopen.99433*

risk of prosthetic joint infection [64]. A large RCT performed within a hip and knee arthroplasty ERAS program showed that increasing a catheterisation threshold from 500 ml to 800 ml over halved the incidence of catheterisation without increase in urological complications [64]. Routine use of urinary catheters is not recommended as per ERAS consensus statement for hip and knee arthroplasty and if used should be removed within 24 hours of insertion [8]. The catheter adds another line or attachment to the patient which increases risk of fall and does not facilitate early mobilisation. Lines should be removed as soon as possible to improve patients' psychological state and avoid any impediment to mobilisation. This not only pertains to urinary catheters but also applies to PCA and other IV lines unless required.


2. Unsteady aerodynamic force

This section is focused on the illumination of unsteady aerodynamic forces (motion-induced forces), which result from the wind-structure interaction. Fluctuating deflections of the structure may be excited by the turbulence in oncoming flow, or the wake instability caused by vortex shedding in the structural wake. The unsteady aerodynamic forces result from the modification of the flow as the structure vibrates or changes shape, in other words, the interaction of the wind flow and structure. These forces may lead to instability. The unsteady aerodynamic force is described as two components: the aerodynamic stiffness term that is in-phase with the displacement and the aerodynamic damping term that is out-phase with displacement.

The aerodynamic stiffness is the added stiffness of the air surrounding the structure, which may increase or effectively reduce available structural static stiffness. For a conventional heavy structure, the aerodynamic stiffness is generally insignificant in comparison to the structural stiffness. However, for a long-span light-weight structure, which vibrates more easily in the wind, the aerodynamic stiffness may change the structural response. For instance, if the total static stiffness of the system in wind is reduced to zero, then a divergent instability may be induced.

When a structure is vibrating in the wind, the relative velocity of the structure to the wind flow changes in magnitude and direction. This phenomenon effectively produces an added damping force, referred to as aerodynamic damping. The aerodynamic damping may add to the structural damping to reduce the response of structure, or become negative and increase the response of the structure. The chances of aerodynamic instability are high as the total damping in the system approaches zero.

#### 2.1. Definition of unsteady aerodynamic force

The displacement of structure in the jth mode may be represented by the following equation,

$$Z\_j(\mathbf{s}, t) = \varphi\_j(\mathbf{s})\mathbf{x}\_j(t) \tag{1}$$

where φj and xj are the mode shape and generalized displacement of the jth mode, respectively; and s represents the circumferential coordinate taken along the roof.

Applying a modal analysis to the equation of motion for the roof, we obtain the following equation of motion for the jth generalized displacement,

$$\mathbf{M}\_{\mathbf{s}\_{\uparrow}}\ddot{\mathbf{x}}\_{j}(t) + \mathbf{C}\_{\mathbf{s}\_{\uparrow}}\dot{\mathbf{x}}\_{j}(t) + \mathbf{K}\_{\mathbf{s}\_{\uparrow}}\mathbf{x}\_{j}(t) = F\_{W\_{j}}(t) + F\_{A\_{j}}(t) \tag{2}$$

$$
\ddot{\mathbf{x}}\_{\dot{\jmath}}(t) + 2\xi\_{s\_{\dot{\jmath}}}\omega\_{s\_{\dot{\jmath}}}\dot{\mathbf{x}}\_{\dot{\jmath}}(t) + \omega\_{s\_{\dot{\jmath}}}^2 \mathbf{x}\_{\dot{\jmath}}(t) = F\_{\dot{\jmath}}(t)/M\_{s\_{\dot{\jmath}}}\tag{3}
$$

$$F\_j(t) = F\_{W\_j}(t) + F\_{A\_j}(\mathbf{x}, \dot{\mathbf{x}}, \ddot{\mathbf{x}} \dots) \tag{4}$$

where Msj = generalized mass, ωsj = natural circular frequency, ζsj = critical damping ratio, and Fj = generalized force. FWj represents the fluctuating wind force due to the oncoming flow and wake instability, while FAj represents the unsteady aerodynamic force due to the wind-roof interaction.

In the case of the forced-vibration test, a steady vibration in the first anti-symmetric mode represented by a sine curve is applied to the roof. The unsteady aerodynamic force FAj (here j = 1) can be obtained from Eq. (5) by using the Fourier series at the frequency fm of the forced vibration:

$$F\_{A\_{\rangle}}(t) = F\_{\mathbb{R}\_{\rangle}} \cos 2\pi f\_m t - F\_{I\_{\rangle}} \sin 2\pi f\_m t \tag{5}$$

$$F\_{R\_{\dot{\gamma}}} = \frac{1}{T} \int\_{-T}^{T} F\_{\dot{\gamma}}(t) \cos 2\pi f\_m t \tag{6}$$

$$F\_{I\_j} = \frac{1}{T} \int\_{-T}^{T} F\_j(t) \sin 2\pi f\_m t \tag{7}$$

where FRj and FIj are the in-phase and out-of phase components of the unsteady aerodynamic force, respectively.

#### 3. Large eddy simulation

Ohkuma et al. [9] investigated the mechanism of aeroelastic instability of long-span flat roofs using a forced vibration test in a wind tunnel. At present, long-span curved roofs are universally constructed. However, there is an insufficient research on unsteady aerodynamic forces on long-span curved roofs, and the characteristics of unsteady aerodynamic forces are not well understood. Therefore, it is necessary to investigate this problem further for proposing more

In this chapter, we investigate the characteristics of unsteady aerodynamic forces acting on long-span curved roofs for improving the wind-resistant design method. The large eddy simulation (LES) is used to discuss the influences of a roof's vibration on the wind pressure and flow field around a vibrating roof. The characteristics of unsteady aerodynamic forces in a wider range of reduced frequency of vibration are also investigated. The results of LES are

This section is focused on the illumination of unsteady aerodynamic forces (motion-induced forces), which result from the wind-structure interaction. Fluctuating deflections of the structure may be excited by the turbulence in oncoming flow, or the wake instability caused by vortex shedding in the structural wake. The unsteady aerodynamic forces result from the modification of the flow as the structure vibrates or changes shape, in other words, the interaction of the wind flow and structure. These forces may lead to instability. The unsteady aerodynamic force is described as two components: the aerodynamic stiffness term that is in-phase with the displacement and the aerodynamic damping term that is out-phase with displacement. The aerodynamic stiffness is the added stiffness of the air surrounding the structure, which may increase or effectively reduce available structural static stiffness. For a conventional heavy structure, the aerodynamic stiffness is generally insignificant in comparison to the structural stiffness. However, for a long-span light-weight structure, which vibrates more easily in the wind, the aerodynamic stiffness may change the structural response. For instance, if the total static stiffness

of the system in wind is reduced to zero, then a divergent instability may be induced.

When a structure is vibrating in the wind, the relative velocity of the structure to the wind flow changes in magnitude and direction. This phenomenon effectively produces an added damping force, referred to as aerodynamic damping. The aerodynamic damping may add to the structural damping to reduce the response of structure, or become negative and increase the response of the structure. The chances of aerodynamic instability are high as the total

The displacement of structure in the jth mode may be represented by the following equation,

where φj and xj are the mode shape and generalized displacement of the jth mode, respectively;

ð Þs xjð Þt (1)

Zjð Þ¼ s; t φj

and s represents the circumferential coordinate taken along the roof.

reasonable methods of response analysis for these roofs.

validated by comparing with the experimental results.

2. Unsteady aerodynamic force

102 Flight Physics - Models, Techniques and Technologies

damping in the system approaches zero.

2.1. Definition of unsteady aerodynamic force

The LES is used to investigate the characteristics of unsteady aerodynamic forces. The influences of a roof's vibration on the wind pressure and flow field around a vibrating roof are also investigated. The simulation is carried out by using a CFD software 'STAR-CD'.

#### 3.1. Computational outline

#### 3.1.1. Computational model

The computational model used in the 'STAR-CD' is shown in Figure 1. In order to investigate the effect of geometric shape on the unsteady aerodynamic force, the rise/span ratio r/L of computational models is assumed to be 0.15, 0.20, and 0.25. The curved roof model is forced to vibrate in the first anti-symmetric mode as shown in Figure 1.

#### 3.1.2. Computational parameters

Table 1 summarizes the computational parameters. In order to discuss the effect of geometric shape on wind-roof interaction, the rise/span ratio is changed from 0.15 to 0.25. The amplitude

Figure 1. Computational model.

Table 1. Parameter of CFD simulation.

x0 of vibration is fixed to 4.0 mm (i.e. x0/L = 1/100). In this study, based on the assumptions of the mean roof height for real structure H\_r = 20 m; the wind speed at mean roof height UH\_r = 20–40 m/s; the natural frequency fs = 0.4–2.5 Hz. We calculated the reduced frequency for real roof f\_r \* = 0.2–2.5, as shown in Table 3. In order to satisfy the similarity principle of real long-span roofs fm \* = f\_r \* (fm \* = the reduced frequency for model), the forced vibration frequency fm should be set at 12.5–156.25 Hz, as shown in Table 2. With regard to the limitation of forced vibration equipment used in the wind tunnel experiment [10], the forced vibration

Table 2. Determination of forced vibration frequency.

frequency cannot be set as large as this. Therefore, it is necessary to use LES to examine the characteristics of unsteady aerodynamic forces in a wider range of reduced frequency. For the LES, we change the forced vibration frequency from 0 to 160 Hz and the range of reduced frequency of vibration is from 0 to 2.5, as shown in Table 1.

#### 3.1.3. Computational domain

Figure 2 shows the computational domain. In this study, the length of span direction equals the span of roof to generate two-dimensional flow that is corresponded with that used in the wind tunnel experiment.

#### 3.1.4. Computational mesh

x0 of vibration is fixed to 4.0 mm (i.e. x0/L = 1/100). In this study, based on the assumptions of the mean roof height for real structure H\_r = 20 m; the wind speed at mean roof height UH\_r = 20–40 m/s; the natural frequency fs = 0.4–2.5 Hz. We calculated the reduced frequency

Forced vibration frequency (fm) 0–160 Hz (10 Hz increment)

) 0–2.5

Wind speed 5 m/s Forced vibration amplitude 4 mm

Rise/span ratio (r/L) 0.15, 0.20, 0.25

quency fm should be set at 12.5–156.25 Hz, as shown in Table 2. With regard to the limitation of forced vibration equipment used in the wind tunnel experiment [10], the forced vibration

*H*\_real

Mean roof height for real structure H\_r = 20 m Mean roof height for model H = 0.08 m Wind speed at mean roof height UH\_r = 20–40 m/s Wind speed at mean roof height UH = 5 m/s Natural frequency fs = 0.4–2.5 Hz Forced vibration frequency fm = 12.5–156.25 Hz

∗ m ¼ f m H UH

\* = 0.2–2.5 Reduced frequency for model fm

\* = 0.2–2.5, as shown in Table 3. In order to satisfy the similarity principle of real

*UH*\_model

\* = the reduced frequency for model), the forced vibration fre-

*fm*䠖Forced vibration frequency

䛆 CFD Simulation 䛇

\* =f\_r

\* = 0.2–2.5

*H*\_model

for real roof f\_r

*UH*\_real

f ∗ \_r ¼ f sH\_ r

long-span roofs fm

Figure 1. Computational model.

104 Flight Physics - Models, Techniques and Technologies

Reduced vibration frequency(fm

Table 1. Parameter of CFD simulation.

\* = f\_r

*f***s** 䠖Natural frequency

UH\_ r f

䛆 Real Structure 䛇

Table 2. Determination of forced vibration frequency.

Reduced frequency for real roof f\_r

\* (fm

\*

In the simulation, various types of mesh arrangements were calculated. We compared the results of LES with those of wind tunnel experiment. And then, the mesh arrangement was selected which leads to the most corresponding results with that of experiment, as shown in Figure 3. The magnitude of minimum mesh is 0.15 103 . And the dynamic mesh is used to simulate the vibration of model.

Figure 2. Computational domain.

Figure 3. Mesh arrangement around roof.

#### 3.1.5. Computational and boundary conditions

There are mainly three types of CFD approaches, which are used in computational wind engineering (CWE): the Reynolds-averaged Navier-Stokes (RANS), the large eddy simulations (LES), and direct numerical simulation (DNS). Due to the limitation of available computer memory and speed at present, DNS cannot be widely used in CWE for solving complicated practical problems. RANS solves the time-averaged NS equations, and the averaged solution reflects the averaged properties of the turbulent flow. Thus, the time-averaged solution is less trustable in nonstationary flows. On the other hand, LES resolves the scale of motion larger than the gird size and the effect of motion of turbulent eddy smaller than grid scale needs to be modeled. The unsteady motions of large eddy can be explicitly predicted and the accuracy is usually much better than RANS models, since the effects of only small eddy are modeled. Therefore, the LES is adopted in this study.

The governing equations adopted in the present LES method are the spatially filtered continuity and Navier-Stokes equations as follows,

$$\frac{\partial \overline{u}\_i}{\partial x\_i} = 0 \tag{8}$$

$$\frac{\partial \overline{u}\_i}{\partial t} + \frac{\partial \overline{u}\_i \overline{u}\_j}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial \overline{p}}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \nu \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} \right) - \frac{\partial \tau\_{ij}}{\partial \mathbf{x}\_i} \tag{9}$$

$$
\pi\_{i\bar{j}} = \overline{u\_i u\_{\bar{j}}} - \overline{u\_i} \overline{u\_{\bar{j}}} \tag{10}
$$

where ui, uj represent flow velocity in i-direction and j-direction. The p, ρ, and υ represent pressure, density, and dynamic viscosity of the fluid, respectively. The (u) denotes application of the spatial filter. The τij is subgrid-scale (SGS) stress, which is parameterized by an eddy viscosity model. The standard Smagorinsky model is adopted to estimate the term of τij, as shown in Eqs. (11)–(13).

$$\tau\_{i\dot{j}} = -2\nu\_{SGS}\overline{S}\_{i\dot{j}} + \frac{1}{3}\delta\_{\dot{j}}R\_{kk} \tag{11}$$

$$\overline{S}\_{i\dot{\jmath}} = \frac{1}{2} \left( \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_{\dot{\jmath}}} + \frac{\partial \overline{u}\_{\dot{\jmath}}}{\partial \mathbf{x}\_i} \right) \tag{12}$$

$$\nu\_{\rm SGS} = \left(\mathbb{C}\_{\rm s}\overline{\Delta}\right)^{2} \sqrt{2\overline{\mathbb{S}\_{\rm ij}\overline{\mathbb{S}\_{ij}}}} \tag{13}$$

where υSGS is the subgrid-scale turbulent eddy viscosity. Sij is the rate of strain tensor for the resolved scale. Δ means the spatial filter width and Cs is the Smagorinsky constant and is taken as 0.12 (Cs = 0.12). The computational and boundary conditions are summarized in Table 3.

Table 3. Computational and boundary conditions.

#### 3.1.6. Inflow turbulence

3.1.5. Computational and boundary conditions

106 Flight Physics - Models, Techniques and Technologies

Therefore, the LES is adopted in this study.

ity and Navier-Stokes equations as follows,

shown in Eqs. (11)–(13).

∂ui ∂t þ ∂uiuj ∂xj

There are mainly three types of CFD approaches, which are used in computational wind engineering (CWE): the Reynolds-averaged Navier-Stokes (RANS), the large eddy simulations (LES), and direct numerical simulation (DNS). Due to the limitation of available computer memory and speed at present, DNS cannot be widely used in CWE for solving complicated practical problems. RANS solves the time-averaged NS equations, and the averaged solution reflects the averaged properties of the turbulent flow. Thus, the time-averaged solution is less trustable in nonstationary flows. On the other hand, LES resolves the scale of motion larger than the gird size and the effect of motion of turbulent eddy smaller than grid scale needs to be modeled. The unsteady motions of large eddy can be explicitly predicted and the accuracy is usually much better than RANS models, since the effects of only small eddy are modeled.

The governing equations adopted in the present LES method are the spatially filtered continu-

where ui, uj represent flow velocity in i-direction and j-direction. The p, ρ, and υ represent pressure, density, and dynamic viscosity of the fluid, respectively. The (u) denotes application of the spatial filter. The τij is subgrid-scale (SGS) stress, which is parameterized by an eddy viscosity model. The standard Smagorinsky model is adopted to estimate the term of τij, as

> 1 3

2SijSij q

τij ¼ �2νSGSSij þ

∂ui ∂xj þ ∂uj ∂xi � �

νSGS ¼ CsΔ � �2 ffiffiffiffiffiffiffiffiffiffiffiffi

where υSGS is the subgrid-scale turbulent eddy viscosity. Sij is the rate of strain tensor for the resolved scale. Δ means the spatial filter width and Cs is the Smagorinsky constant and is taken as 0.12 (Cs = 0.12). The computational and boundary conditions are summarized in Table 3.

Sij ¼ 1 2 ¼ 0 (8)

δijRkk (11)

(9)

(12)

(13)

� ∂τij ∂xi

τij ¼ uiuj � uiuj (10)

∂ui ∂xi

¼ � 1 ρ ∂p ∂xi þ ∂ ∂xj ν ∂ui ∂xj � �

As is known, the flow around a structure is strongly affected by the flow turbulence. Therefore, the proper generation of the inflow turbulence for the LES is essential in the determination of wind loads on structures. At present, several techniques have been developed. In general, there are three kinds of inflow turbulence generation methods. The first approach is to store the time history of velocity fluctuations obtained from a preliminary LES computation. Nozu and Tamura [11] employed the interpolation method with the periodic boundary condition to simulate a fully developed turbulent boundary layer and tried to change the turbulent characteristics by using roughness blocks. Another approach is to numerically simulate the turbulent flow in auxiliary computational domains (often called a driver region set at the upstream region of a main computational domain). Lund et al. [12] proposed the method to generate turbulent inflow data for the LES of a spatially developing boundary layer. Kataoka and Mizuno [13] simplified Lund's method by assuming that the boundary layer thickness is constant within the driver region, and only the fluctuating part of velocity is recycled in the streamwise direction. Nozawa and Tamura [14, 15] discussed the potential of large eddy simulation for predicting turbulence characteristics in a spatially developed turbulent boundary layer over a rough ground surface and improved Lund's method. The third approach is to use artificial numerical models to generate inflow turbulence statistically [16–19].

In this study, we use a preliminary LES to simulate inflow turbulence and store the time history of velocity fluctuations. Figure 4 shows a schematic illustration of the domain of the preliminary computation. In the domain, the roughness blocks with heights 3, 5, and 8 cm are distributed on the ground to generate turbulence. The computational and boundary conditions are summarized in Table 4.

Figure 4. Preliminary computational domain.

Table 4. Computational and boundary conditions.

The profiles of the mean wind speed and turbulent intensity at the inlet of the computational domain are shown in Figure 5(a). The longitudinal velocity spectrum at a height of H = 90 mm is shown in Figure 5(b). In both figures, the results of wind tunnel flow are also plotted for comparative purposes. It can be seen that the inflow turbulence used in the LES is generally in good agreement with that used in the wind tunnel experiment.

Figure 5. Comparison of inflow turbulence between wind tunnel experiment and LES; (a) profiles of mean wind speed (U(z)) and turbulence intensity (Iu); (b) longitudinal velocity spectrum (H = 90 mm).

### 4. Results and discussion

#### 4.1. Comparison with wind tunnel experiment

In order to validate the LES computation, the distributions of the mean wind pressure coefficient Cp\_mean and fluctuating wind pressure coefficient Cp\_RMS along the centerline of the vibrating roof is compared with those obtained from the wind tunnel experiment. Figures 6 and 7 show the results, in which the results for the frequencies of 0, 10, and 15 Hz are plotted. It can be seen that there is generally a good agreement between the LES and the wind tunnel experiment. In Figure 6, the difference is somewhat larger near the rooftop; the LES values are approximately 10% larger in magnitude than the experimental ones. This difference may be due to a difference in surface roughness of the roof between the LES and the wind tunnel experiment. In Figure 7, when the fm = 0 Hz, the value of Cp\_RMS for the LES is larger than that for the wind tunnel test. That maybe because that the turbulence intensity of inflow turbulence used in the LES is slightly larger than that used in the wind tunnel test (see Figure 5).

#### 4.2. Distribution of wind pressure on the roof

The profiles of the mean wind speed and turbulent intensity at the inlet of the computational domain are shown in Figure 5(a). The longitudinal velocity spectrum at a height of H = 90 mm is shown in Figure 5(b). In both figures, the results of wind tunnel flow are also plotted for comparative purposes. It can be seen that the inflow turbulence used in the LES is generally in

Upper boundary Zero normal velocity and zero normal gradients of other variables

good agreement with that used in the wind tunnel experiment.

Inlet boundary Cyclic boundary condition

Side boundary Cyclic boundary conditions Outlet boundary Cyclic boundary conditions

Diffusion schemes Centered difference scheme Time differential schemes First order Euler implicit

Floor and surfaces of roughness blocks No-slip condition Convection schemes MARS method

Numerical algorithm PISO algorithm Time step Δt = 2.0E�04 s

Table 4. Computational and boundary conditions.

Figure 4. Preliminary computational domain.

108 Flight Physics - Models, Techniques and Technologies

The distributions of mean and rms fluctuating wind pressure coefficients for various forcedvibration frequencies are shown in Figure 8. It can be seen that the mean wind pressure

Figure 6. Comparisons for the distribution of the mean wind pressure coefficients along the centerline between LES and wind tunnel experiment.

Figure 7. Comparisons for the distribution of the fluctuating wind pressure coefficients along the centerline between LES and wind tunnel experiment.

coefficients Cp\_mean near the rooftop increase in magnitude and the rms fluctuating wind pressure coefficients Cp\_rms generally increase, as the forced-vibration frequency increases. Furthermore, the variation is significant near the position of the greatest forced-vibration amplitude. These results indicate that the wind pressure field around the vibrating roof is strongly influenced by the vibration of the roof.

LES of Unsteady Aerodynamic Forces on a Long-Span Curved Roof http://dx.doi.org/10.5772/intechopen.70880 111

Figure 8. Variation of mean and fluctuating wind pressure coefficients with forced vibration frequency (r/L = 0.15).

Figure 9 shows the variations of mean and rms fluctuating wind pressure coefficients with the rise/span ratio. It can be seen that the Cp\_mean changes from negative to positive at the leading edge of the roof as the rise/span ratio increases. Furthermore, the negative peak value increases in magnitude with an increase in rise/span ratio. The value of Cp\_rms increases with an increase in rise/span ratio at the middle part of the roof. However, the effect of r/L on Cp\_rms is less significant than on Cp\_mean.

#### 4.3. Discussion flow field around the roof

coefficients Cp\_mean near the rooftop increase in magnitude and the rms fluctuating wind pressure coefficients Cp\_rms generally increase, as the forced-vibration frequency increases. Furthermore, the variation is significant near the position of the greatest forced-vibration amplitude. These results indicate that the wind pressure field around the vibrating roof is

Figure 7. Comparisons for the distribution of the fluctuating wind pressure coefficients along the centerline between LES

Figure 6. Comparisons for the distribution of the mean wind pressure coefficients along the centerline between LES and

strongly influenced by the vibration of the roof.

wind tunnel experiment.

110 Flight Physics - Models, Techniques and Technologies

and wind tunnel experiment.

The roof configurations at several steps (phases) during one period of vibration are shown in Figure 10. The deformation of the windward side is upward and becomes the greatest at step 2; and that of the leeward side is upward and becomes the greatest at step 4.

Figure 9. Variation of mean and fluctuating wind pressure coefficient with rise/span ratio (fm = 10 Hz).

Figure 10. Roof configurations at several steps.

Figure 11 shows representative flow fields around a stationary or vibrating roof at a frequency of 10 or 20 Hz. It can be seen that the wind speed increases near the roof regardless of the roof's vibration. In the case of a stationary roof (fm = 0 Hz), the flow separates near the 3/4 position of the roof from the leading edge. On the other hand, in the case of a vibrating roof, the separated vortex seems smaller than that in the stationary roof case, which may be due to the vibration of the roof that restrains the separation of the vortex. In addition, the separated position at the

Figure 11. Flow fields around the roof for various forced vibration frequencies (r/L = 0.15); (a) 0 Hz; (b) 10 Hz; (c) 20 Hz.

rear of roof changes with the vibration of roof. The separated position is relatively forward when the roof is vibrated in step 1 to step 3, because the deformation at the windward side of roof makes the flow separated in advance. On the other hand, the separated position is relatively backward when the roof is vibrated in step 3 to step 5, as the result that the deformation at the leeward side of roof restrains the flow separated.

Figure 12 shows the effect of the rise/span ratio on the flow field around a vibrating roof at a forced vibration frequency of 20 Hz. It can be seen that the wind speed near the rooftop becomes higher, generating larger suction as the rise/span ratio increases. Therefore, the negative peak value of Cp\_mean increases with an increase in rise/span ratio (see Figure 8). Furthermore, as the rise/span ratio increases, the vortex at the rearward of roof becomes larger. Flow fields around the roof for various rise/span ratios (fm = 20 Hz); (a) r/L = 0.15; (b) r/L = 0.20; (c) r/L = 0.25.

#### 4.4. Evaluation of unsteady aerodynamic forces

Figure 11 shows representative flow fields around a stationary or vibrating roof at a frequency of 10 or 20 Hz. It can be seen that the wind speed increases near the roof regardless of the roof's vibration. In the case of a stationary roof (fm = 0 Hz), the flow separates near the 3/4 position of the roof from the leading edge. On the other hand, in the case of a vibrating roof, the separated vortex seems smaller than that in the stationary roof case, which may be due to the vibration of the roof that restrains the separation of the vortex. In addition, the separated position at the

Figure 9. Variation of mean and fluctuating wind pressure coefficient with rise/span ratio (fm = 10 Hz).

Figure 10. Roof configurations at several steps.

112 Flight Physics - Models, Techniques and Technologies

In this study, we use aerodynamic stiffness coefficient aKj and aerodynamic damping coefficient aCj to investigate the characteristics of unsteady aerodynamic forces acting on a vibrating long-span curved roof, which are given by the following equations [8]:

Figure 12. Flow fields around the roof for various rise/span ratios (fm = 20 Hz); (a) r/L = 0.15; (b) r/L = 0.20; (c) r/L = 0.25.

$$a\_{kj} = \frac{F\_{Rj}(f\_m)}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} = \frac{1}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} \frac{1}{T} \int\_{-T}^{T} F\_j(t) \cos 2\pi f\_m t \tag{14}$$

$$a\_{\mathbb{C}\bar{\jmath}} = \frac{F\_{\bar{\jmath}}\left(f\_m\right)}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} = \frac{1}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} \frac{1}{T} \int\_{-T}^{T} F\_{\bar{\jmath}}(t) \sin 2\pi f\_m t \tag{15}$$

where FRj is the in-phase component with the generalized displacement represented as the aerodynamic stiffness term, FIj is the in-phase component with velocity represented as the aerodynamic damping term, qH = velocity pressure at the mean roof height H, As = roof area, x0 = forced vibration amplitude, L = span of the roof, T = vibration period, fm = forced vibration frequency, and fm \* = reduced frequency of vibration, defined by fmH/UH, with UH being the mean wind speed at the mean roof height H.

The generalized force Fj may be described in terms of the external and internal pressures pe and pi, as shown in Eq. 16,

$$F\_{\dot{\!\!\!/}}(t) = \int\_{0}^{R\_{\text{s}}} \left[ p\_{e\_{\dot{\!\!/}}}(s, t) - p\_{i\_{\dot{\!\!/}}}(s, t) \right] \wp\_{\dot{\!\!/}}(s) ds \tag{16}$$

where Rs = total length of the vaulted roof. Internal pressure pi is ignored in the present study, because the first anti-symmetric mode under consideration causes no change of internal volume. The model's vibration mode almost corresponded with the asymmetric sine mode, as shown in Eq. 17.

$$\varphi\_j(\mathbf{s}) = \sin 2\pi \frac{\mathbf{s}}{R\_s} \tag{17}$$

Figure 13 shows the aerodynamic stiffness and damping coefficients, aK and aC, obtained from the LES and the wind tunnel experiment, plotted as a function of the reduced frequency of vibration fm \* (fm \* = fmH/UH). The wind tunnel experiment was carried out in a limited range of fm \* , while the LES was conducted over a wider range of fm \* . It can be seen that the LES results are consistent with those of the wind tunnel experiment, which indicates that the LES model can be used for investigating the characteristics of unsteady aerodynamic forces. The aerodynamic stiffness coefficient aK is generally positive and increases with an increase in fm \* , which decreases the total stiffness of the system. On the other hand, the aerodynamic damping coefficient aC is negative and increases in magnitude with an increase in fm \* , resulting in an increase in the total damping of the system.

The distribution of aerodynamic stiffness and damping coefficients aK and aC with fm \* for various rise/span ratios is shown in Figure 14. It can be seen that the values of aK for r/L = 0.15, 0.20, and 0.25 are generally consistent with each other when fm \* <0.4. However, when fm \* >0.4, the value of aK decreases with an increase in the rise/span ratio. Regarding the value of aC, the results for various rise/span ratios are generally similar to each other. This figure indicates that the value of aK is influenced by the rise/span ratio of a long-span vaulted roof. However, the effect of the rise/span ratio on the value of aC is small.

aKj ¼ FRj f m

114 Flight Physics - Models, Techniques and Technologies

aCj ¼ FIj f m

mean wind speed at the mean roof height H.

frequency, and fm

pi, as shown in Eq. 16,

� � qHAsð Þ x0=L ¼ 1

� � qHAsð Þ x0=L ¼ 1

FjðÞ¼ t

ðRs 0 pej

qHAsð Þ x0=L

Figure 12. Flow fields around the roof for various rise/span ratios (fm = 20 Hz); (a) r/L = 0.15; (b) r/L = 0.20; (c) r/L = 0.25.

qHAsð Þ x0=L

where FRj is the in-phase component with the generalized displacement represented as the aerodynamic stiffness term, FIj is the in-phase component with velocity represented as the aerodynamic damping term, qH = velocity pressure at the mean roof height H, As = roof area, x0 = forced vibration amplitude, L = span of the roof, T = vibration period, fm = forced vibration

The generalized force Fj may be described in terms of the external and internal pressures pe and

ð Þ� s; t pij

h i

1 T ðT �T

1 T ðT �T

\* = reduced frequency of vibration, defined by fmH/UH, with UH being the

ð Þ s; t

φj

Fjð Þt cos 2πf mt (14)

Fjð Þt sin 2πf mt (15)

ð Þs ds (16)

Figure 13. Comparisons of the LES and the wind tunnel experiment for the aerodynamic stiffness coefficient aK and aerodynamic damping coefficient aC versus reduced frequency of vibration fm \* .

Figure 14. Aerodynamic stiffness and damping coefficients versus fm \* for different rise/span ratio r/L.


**5. Conclusions**

In this paper, we have created a simulated asset market and modeled common AT strategies. We conducted two experiments in this market. Our contribution is the creation of a platform capable of studying the impact of AT trading on market quality directly. This expands our capability of evaluating AT beyond conventional empirical studies. It provides a method of cross-checking empirical findings, gaining deeper insights into them, and also potentially forming new hypotheses for further empirical research.

In Experiment 1, we investigated the impact of AT on market liquidity. We simulated a gradual rise in the participation of AT in a market with fundamental and technical traders of both AT and human types. We found that liquidity rose sharply as AT was introduced with most of the liquidity rise realized at just 10% AT participation. This leads to an interesting hypothesis for future empirical work to see if there is such an early threshold for reaping the liquidity benefits of AT. This could hold important implications for developing countries with nascent AT participation.

In Experiment 2 we investigated the impact of AT on price discovery. We simulated a rise in algorithmic pairs trading that apply statistical arbitrage to make profits based on short-term price correlations. In this experiment, we found high price correlations between the pair traded, even when fundamental values had very little correlation. This supports some findings in the empirical literature that indicate AT leading to price movements away from fundamentals.

**Author Contributions:** Conceptualization, P.M.; Methodology, P.M. and C.C.; Software, C.C., T.W., and B.X.; Validation, P.M., C.C., T.W. and B.X.; Formal Analysis, P.M., C.C., T.W. and B.X.; Investigation, P.M., C.C., T.W. and B.X.; Resources, P.M., and C.C.; Data Curation, P.M., C.C., T.W. and B.X.; Writing-Original Draft Preparation, P.M., C.C., T.W. and B.X.; Writing-Review & Editing, P.M., C.C. and T.W.; Visualization, P.M., C.C., T.W. and B.X.; Supervision, P.M., and C.C.; Project Administration, P.M. and C.C.; Funding Acquisition, P.M. and C.C.

**Funding:** This research was funded by research grants at Connecticut College.

**Conflicts of Interest:** The authors declare no conflict of interest.