Simulation of Ground Power Unit-3 Stirling Engine With Air as Working Fluid

Around 80% of the energy used worldwide comes from fossil fuels [1]. A large portion of the current global energy system is unsustainable; thus, switching to renewable energy sources is probably the best course of action for both the environment and humanity [2,3]. Around the world, three-quarters of the additional renewable capacity in 2023 came from solar PV alone. By 2028, solar PV and wind additions are expected to have more than doubled from 2022 levels [4]. To contribute to this addition, several hybrid solutions, such as grid-connected renewable energy systems with or without energy storage or combined heat and power systems, are being researched [5,6].

With over 200 years of existence, the Stirling engine has proven to be versatile in a variety of settings. Because they only have a few moving parts, they can have a simple design, little vibration and noise, and a high efficiency of roughly 35%. Renewable energy sources like solar, waste heat, or biomass can also be used to power Stirling engines [7,8]. Stirling engines powered by solar energy and hybrid systems have been examined by many authors [9–13]. Common applications being explored for solar-based Stirling engine systems include water distillation and desalination, water pumping, hybridization and storage, off-grid electrification, combined heat and power, and water distillation. Other uses of the Stirling cycle include heat pumps and cryocoolers, used in cooling gases to cryogenic temperatures [14].

The β-, γ-, and α-type engines are the three types of Stirling engines [15]. Free-piston engines are Stirling engines that operate without a piston driving mechanism [16]. One energy system whose operation requires a thorough thermodynamic analysis is the Stirling engine [17]. This type of research frequently calls for a complex model. The performance of Stirling engines is designed and predicted using five different strategies: zero-order, first-order, second-order, third-order (also known as the nodal design approach), and fourth-order (also known as computational fluid dynamics (CFD) simulation). In this sense, a higher order indicates a more complicated model; “order” refers to classification according to the complexity level of the model [17,18].

Every loss is disregarded in the first-order analysis. Because most of the irreversibilities that arise during Stirling engine running are not taken into account, this type of analysis is extremely straightforward yet unreliable. First-order analysis examples include using the Baele number or Iwamoto's more advanced method, which includes experiment findings for performance prediction [19,20]. In second-order models, calculations are done first on the premise that there are no losses. Upon reaching convergence in the solution, different kinds of losses are subsequently evaluated separately. These include heat exchanger losses, conduction and radiation losses, pressure drop, losses in the appendix gap, and inefficiencies in regenerator thermal inefficiency [21,22]. Pressure drop, heat exchanger losses, appendix gap losses, conduction and radiation losses, and regenerator losses are some of these losses.

A second-order model known as polytropic analysis of Stirling engine with varying losses was published by Babaelahi and Sayyaadi [21]. The adiabatic compression and expansion mechanisms employed in the earlier models were replaced by polytropic ones in this one. Building upon the Simple model, Yang et al. recently presented an improved second-order adiabatic model known as the Incorporated Pressure Drop-modified Simple Model. Compared to previous adiabatic models such as the Improved Simple Analytical Model and the Combined Adiabatic-Finite Speed, it demonstrated superior performance prediction at high pressure and frequency conditions [23].

Third-order techniques discretize the geometry of the engine into a network of nodes and control volumes and solve a system of differential equations [24]. When compared to second-order models, the analysis of third-order models is typically more complex and requires more time and computing power [25]. Aksoy et al. [26] separated the overall volume of the Stirling engine into 22 nodal volumes using third-order analysis to analyze and contrast the functioning of the rhombic and crank drive mechanisms.

The first successful three-dimensional (3D) CFD simulation of a biomass-powered γ-type Stirling engine was carried out in 2006 by Mahkamov [27]. In comparison to the second-order outcomes, which were 30% off from the experimental data, the forecasts were just 12–18% off. Remeshing was achieved by adding or removing cell layers above the power and displacer pistons during the stroke, using the k–ε turbulence model. Due to a significant pressure drop in the regenerator, a “dead” zone in the compression space, and gas entrapment in the connecting pipe between the two portions of the compression space, the original engine experienced reduced power output. The engine was then converted from gamma to alpha type to improve power output.

El-Ghafour et al. [28] recently carried out a three-dimensional simulation in CFD on the Ground Power Unit-3 (GPU-3) Stirling engine using a detailed geometry of the genuine engine. After comparing six turbulent models that simulated the engine with the experimental data, the most accurate result, with a difference of only 4%, was found to be the realizable enhanced wall treatment (EWT) k–ε model. Engine performance is influenced by the working fluid; hydrogen performs better than helium due to its smaller molecular weight, stronger thermal conductivity, and lower specific heat capacity. There is also a representation of the simulated versus experimental data, with the realizable k–ε EWT model being in good agreement with the experimental data.

Using a verified 1D third-order model, Rogdakis et al. [29] validated a three-dimensional CFD model for a β-type Stirling engine. While not exact, the simulated engine employed in the study is comparable to the GPU-3 Stirling engine. A power output of 6.93 kW and an efficiency of 37% were anticipated by the model. A CFD parametric research was conducted on a 100-W Stirling engine by Cheng and Phung [30], and the results were verified through an experiment. Unlike the prior traditional k–ε model, they changed the piston's length, heating and cooling temperatures, charge pressure, and rotation speed. The results of the simulation show that there is a direct relationship between the indicated power (or charged pressure) and the overall mass.

Therefore, by identifying complex gas flow behavior as well as unequal pressure and temperature distributions within the engine, a CFD simulation aids in a better understanding and optimization of the engine's design parameters. It also allows for the identification of vortices and the impinging effect, which enhances heat transfer within the engine [7].

Air, hydrogen, and helium are common Stirling engine operating fluids. Helium and hydrogen are the two working fluids that provide the engine with the most performance. Hydrogen is the most efficient but highly combustible and challenging to seal, and helium is scarce on Earth. Air is the least expensive and safest operating fluid, but its performance in the engine is not as good as that of its counterparts. Studies that analyze the performance of the Stirling engine utilizing 3D CFD are limited, with even limited studies being performed with air as a working fluid. A 3D CFD study that evaluates the power output of the Stirling engine with the only variable being air and helium as working fluids was not present in recent literature.

In this work, a 3D CFD analysis was performed of the GPU-3 Stirling engine with air as the working fluid using ansys fluent. GPU-3 engine was utilized because the engine dimensions and experimental data are readily available. The eddy viscosity model used was realizable EWT k–ε model because of its accuracy [28] and validated with experimental data [31]. The power output was compared with helium as working fluid under similar conditions. The temperature, pressure, and volume variation against one revolution of the crank was also evaluated.

The GPU-3 engine is a rhombic drive, single-cylinder, β-type design Stirling engine with sliding rod seals. General Motors Research Laboratories initially constructed this engine in 1965 for the American Army. Later, it was transformed into a research configuration by the NASA Lewis Research Centre [31].

The cylinder, regenerator, cooler, cooler-end connector, power piston, displacer piston, heater, and rhombic driving mechanism are the engine's primary parts. Within the cylinder, the power and displacement pistons are positioned concentrically. Since the displacer is attached to a rod that travels through the piston, it moves up and down using the same drive mechanism as the piston, but at a predetermined phase angle. The displacer creates the compression and expansion areas inside the cylinder. The cylinder is fixed with the cooler, regenerator, and heater, see Fig. 1 which shows the one-eighth of the engine's fluid domain.

The heater is equipped with a single row of 80 circumferential heater tubes. These tubes are divided into two sets, as seen in Fig. 2, and are arranged alternately with a small gap between them. These two sets of tubes are joined by a heater header and a horizontal annular manifold. One set of tube ends in the regenerator intake and the other set ends in expansion space. The regenerator's gas ascends these tubes, passes through the common annular header, and enters the tubes, which point down and are soldered, into the top of the cylinder, also known as hot space.

The regenerator is composed of eight cartridges placed circumferentially around the cylinder. A wire mesh screen matrix is layered above each other in each regenerator, Stainless steel is used to create this matrix.

The cooler is made up of 312 circular tubes that are arranged in a shell-and-tube pattern. Below each regenerator cartridge, 39 cooler tubes are grouped and water is used as coolant. Each cooler's end empties into a cooler-end cap into a duct, known as the cooler-end connecting duct, which is joined to the bottom of the cylinder, also known as cold space. Stainless steel, which has a thermal conductivity of 15.5 W/m K, is used throughout the engine [31]. Geometrical data for the GPU-3 engine are given in Table 1.

Here, a unique mechanism for the β-type Stirling engine, the rhombic drive mechanism is employed. Two cranks are connected by the displacer and power piston connecting rods (links), and they are geared in tandem to drive the yokes by counter-rotating. The symmetrical rhombic drive mechanism works by decreasing the coaxial motion of the piston and displacer, hence minimizing the lateral force acting perpendicular to the axis. This method does not cause any sinusoidal fluctuations in volume. The precise functions of expansion and compression volumes are found using the geometrical properties of the cylinder and the kinematics of the driving mechanism.

where R_c is the crank radius, L is the length of connecting rod (link), e is eccentricity, L_p is the length of the piston, L_pcr is the length of piston rod, L_d is the length of the displacer, and L_dcr is the length of the displacer rod. The position of the top and bottom of the displacer and the top of the power piston against the crank angle is shown in Fig. 3.

In this investigation, the flow was considered viscous, Newtonian, compressible, and oscillatory.

Using a Cartesian coordinate system, the unsteady Reynolds-averaged Navier–Stokes equations can be constructed in tensor form by dividing the flow parameters into group-averaged values (u, T, and p) and the corresponding fluctuations (u′, T′, and p′).

The regenerator is represented utilizing the porous media approach. The two main physical phenomena that must be evaluated while characterizing the regenerator are heat transfer and flow friction.

where $μf$ is the working fluid viscosity

The regenerator's heat flow is modeled using the local thermal nonequilibrium method. A dual-cell approach is applied for both fluid flow and solid matrix. Additional inputs are necessary to account for the energy exchange between the two stages. The fluid and matrix energy equations contain this source term [33].

Work and heat equations:

Figures 1 and 2 show that the geometry of the engine is exactly symmetrical about the cylinder axis. This simulation only employs one-eighth of engine geometry as the computational domain in order to decrease computational expenses. The domain also comprises regenerator, cooler, appendix gap, heater, and cooler-end connecting duct as well to the displacer cylinder. Apart from the displacer, none of the component walls are taken into consideration.

Figure 5 shows the computational domain's configuration as expressed by the computational grid. The engine's initial configuration is shown in the figure when the crank angle is equal to 0 deg. Using SolidWorks v2020 and Mesh (fluent), respectively, the domain's computer aided design (CAD) geometry and grid are produced. The computational domain was divided into 13 different subvolumes with different topologies because of its complex structure. Each of them is meshing individually. The subvolumes are connected by faces of adjacent volumes, which are connected by arbitrary interfaces. Depending on the characteristics of the nearby subvolume, either a sliding or static interface was used, which include static zones, rigid body motion, and deforming.

To enhance the accuracy and convergence of the solution, transitional and low-Re turbulence models were used during the meshing phase, by adhering to the following criteria.

• Hexahedral elements were used for the majority of the grid's construction, with the remaining portion being made of tetrahedral elements. It is more appropriate in this scenario since it can reduce the requirement for computational memory and the risk of numerical diffusion [33,34].

• Locations where substantial gradients of flow variables are anticipated were carefully refined, such as near-wall boundary areas, locations of area change, and areas surrounding abrupt curves and corners. In particular, as many cells as feasible with y + 1 cover the boundary layer in the normal direction [33,36]. All engine walls – apart from those that move – have this refining. This is explained by the fact that moving borders necessitates an adaptive refining method, which significantly lengthens the computation time.

• The mesh was carefully condensed on both sides at the fluid–solid interface to capture the conjugate heat transfer physics at the displacer walls. Two spatially contiguous grids were used for the regenerator's unique topology, which is comprised of two contiguous zones – one for the fluid matrix and the other for the solid [34]. To precisely depict the impact of flow jets entering the regenerator, the grids at both ends of the regenerator were improved.

• While going from small to large things, the grid cell sizes did not change abruptly. Large elements may be found in low-interest locations, to minimize the total number of elements [34].

Eighths of the physical domain were simulated, and the faces of this sector were subjected to periodic azimuthal boundary conditions. Nonslip boundary criteria were assigned to every wall, both moving and stationary. The walls inside the displacer were presumed to be adiabatic, while the walls inside the piston were assumed to be kept at 305 K. As shown in Table 2, the operating condition does not specify the heater tube wall temperature explicitly. Therefore, this temperature was obtained through iteration in order to satisfy two criteria. First, it produces approximately 977 K as a mean cyclic gas temperature in cylinders. Second, the heat supplied to the engine should be constrained to the length of the tubes exposed to the heat source, while the remainder of the tubes should maintain a virtually zero cyclic heat transfer rate [28].

In the same way, lower wall temperatures were enforced. For expansion and compression spaces, the assumption of boundary temperature of an isothermal wall was made possible by the solid material's greater thermal inertia compared to the gas. The cyclic gas average temperature inside each compartment was close to the assumed wall temperature. The graphical representation and values for the heater, cooler, expansion, and compression space wall's assumed temperatures used in the current simulation are shown in Fig. 6 and Table 2. The walls of the remaining components show a linear distribution of temperature.

After multiple iterations, the initial pressure is chosen to be marginally higher than the cycle's minimal pressure, the cycle begins at θ = 0 deg with an initial pressure of 3.8 MPa. The crank speed for the engine is 2000 rpm, with a mean pressure of 4.14 MPa and an eddy viscosity model of realizable k–ε EWT.

fluent, which is a finite volume-based solver was used, and the governing equations were discretized and sequentially solved. A segregated SIMPLEC solver which is transient and pressure-based with an absolute velocity formulation has been used to solve the simulation. The thermophysical properties of the working gas, such as specific heat, viscosity, and thermal conductivity, vary by temperature. The least squares method was used to determine transport variable gradients on the cell border faces. Additional spatial discretization methods applied in simulations include standard for the pressure term and second-order upwind for continuity, momentum, energy, and transport equations of flow models. In terms of density, energy, and turbulence quantities, 0.9 was the under-relaxation factor. The discretization of the temporal derivatives was done using the first-order implicit formulation. fluent 2022r1 was used with the default parameters for all flow models.

To speed up the convergence of calculations, Simulation was started with a steady-state solution, i.e., heat transfer calculations were incorporated but piston and displacer motion was not considered. The standard k-turbulence model, the absence of conjugate heat transfer, and the local thermal equilibrium porous media approach were necessary to fulfill the convergence criteria in the initial cycle of the transient solution. After this when the movement of the displacer and power piston is considered, it becomes a transient case. The calculation advances steadily toward convergence once the fundamental parameters are enabled and the first cycle reaches convergence.

The piston and displacer's reciprocating action, and therefore, the affecting and deforming volumes, were replicated using dynamic mesh technique. Using velocity equations and the DEFINE_CG_MOTION Macro, user defined functions (UDFs) have been created and linked to fluent to regulate the motion of the displacer and power piston. When compression space is completely compressed or the displacer is at top dead center (TDC) in the expansion space, grid stacking was utilized to prevent skew cells [33].

Heat transfer coefficients and flow resistance, which were experimentally determined by Tew et al. [37] experimental results, are required to be introduced into the porous media model. The GPU-3 regenerator's experimental data was presented as friction factor and Nu as functions of Re. On the basis of the proposed correlation, Eqs. (13) and (21) are used to compute the simulation-constant values of the resistance coefficients, 1/α and C₂. The fluid-to-matrix variable coefficient of heat transfer is defined using a UDF with DEFINE_PROFILE as macro and Eqs. (20) and (21). Table 3 summarizes the resistance coefficients, Nu correlations, and interfacial area density for the GPU-3 engine regenerator. The superficial velocity approach is used to simulate porous media because it is numerically more stable [33,34].

There were two main criteria for the simulation's convergence because of the transient character. For the transport equations including energy, the convergence criteria for each cycle is set to 1 × 10⁻⁶, and for all other equations, the convergence criteria is set to 1 × 10⁻³. Second, to monitor certain quantities to confirm that the cyclic steady-state solution has been reached. As a result, for the course of two cycles, the average fluctuation in gas temperature across the 13 subdomains must be less than 1%. The time step of 8.33 × 10⁻⁵ s was used as it corresponds to 1 deg of crank rotation. Power was determined for three different number of elements were tried, out of which 2.42 × 10⁶ was chosen because of lesser elements without an increase in accuracy, as represented in Table 4.

To obtain a cyclic solution, each simulation takes about six cycles, and on a Core i7-12700 processor, each cycle takes about 36 h with a 2.1-GHz clock speed and dynamic memory of 16 GB.

To validate the model, a comparison between the present simulation's indicated power for He and experimental results of the GPU-3 engine for He as working fluid is summarized in Refs. [28,37]. The comparison is illustrated in Fig. 7; here, the realizable EWT k–ε turbulence model was utilized.

The comparison suggests that the current CFD simulation corresponds closely to the experimental outcome. The simulation overestimated power and efficiency by approximately 5% and 7%, respectively. Interestingly, the variance of predicted power was greater than that of predicted efficiency.

It seems that CFD modeling is more accurate in this case. The current results have inaccuracies because radiation heat transfer between the engine walls, flow leakage through the piston rings from the cylinder to the buffer area, and loss of heat conduction through the thickness of material are not taken into consideration.

Figure 8 shows the instantaneous changes in the volumes of the total, expansion space, and compression space as a function of crank angle over the course of a full cycle. All of the working and dead volume spaces are included in the overall volume. It is evident from a comparison of Figs. 3 and 8 that the engine's highest and smallest total volumes occur at the power piston's bottom dead center (BDC) and TDC.

The fluctuation of average gas pressure in the engine is illustrated in Fig. 9, the mean value was equal to 4.1 MPa. Although pressure drop exists and causes variations in gas pressure between engine areas, it was quite small in comparison to gas pressure. Even if pressure drop had a significant impact on losses, different engine areas do not see a varied trend in pressure variation. Because of this, the gas pressure's magnitude and variance in all engine areas are very descriptive of gas pressure.

The fluctuation in gas temperature is shown in Fig. 10. It is evident that there was a limited temperature variation in the cooler and heater. The heater and cooler's respective temperatures match up with the heat exchanger wall temperatures. The gas in the heater was typically colder than the heater walls during the cycle. Additionally, the cooler's gas was hotter than the walls. Because the temperature of the gas in the heater was lower than the walls, gas absorbs heat from the external heat source.

The gas temperature swings in the cold space (compression space) and the hot space (expansion space) are larger than in the cooler and heater, respectively. The gas in the cold area was hotter than the gas in the cooler over the majority of the cycle, while the gas in the hot space was colder than the gas in the heater. The gas inside the cold area became colder than the gas in the cooler at a point near the lowest pressure. In certain cases, the cooler does not reject heat; rather, it absorbs heat from the heat sink.

As illustrated in Fig. 11, the cycle starts at (θ = 0 deg) when the compression space (cold space) was close to its maximum, i.e., the displacer was close to its top dead center and power piston was close to its bottom dead center. The expansion and compression space are simultaneously contracted as they go upward. In this instance, the majority of gas travels from expansion space to compression space via heat exchangers, regenerator and some gas travels via appendix gap (clearance). This hot gas that jets to compression space results in the mixing of a hot jet from expansion space with the cold gas present in compression space. At θ = 45 deg, the gas was moving through both channels, from compression space to expansion space. The displacers' velocity gradually slows as it reaches its TDC, and gradually, the flow direction reverses to compression space toward expansion space, as the displacer starts going downwards. Nonetheless, a small quantity of heated gas continues to escape from the expansion space via heater pipes. The flow through heater pipes now starts flowing in reverse direction. The displacer allows an increase of the expansion space after descending and achieving TDC.

Around θ = 90 deg, as the displacer's downward acceleration rises, so does the cold flow from the compression space entering the expansion space through the appendix gap and the flow from heater tubes. This accounts for the abrupt decrease in the expansion space's average gas temperature at the beginning of its expansion stroke. As soon as the piston reverses direction and reaches its TDC (θ = 110 deg), the compression space is fully compressed. This results in a significant amount of flow being expelled from compression space to expansion space and the temperature of expansion space increases during the range of θ = 100 deg to 160 deg. Therefore, the engine's overall capacity has increased but the pressure is still rising. A pair of counter-rotate vortices, see Fig. 12, with varying intensities, is evident as the main flow field feature throughout the expansion (downward) stroke of displacer, i.e., till around θ = 225 deg. The expansion stroke has a continuous nature, as shown in Fig. 11, these vortices improve mixing within the area while also significantly lowering the average gas temperature in expansion space.

The flow changes direction again, progressively, from the expansion to the compression space as the displacer approaches its BDC, at around θ = 250 deg. Some amount of gas enters expansion space from compression space via the appendix gap and most of the gas enters via heat exchangers and regenerators. During this time, the gas temperature in compression space is almost constant. The two opposing effects that result in this situation are the cooling effect, as a result of volume expansion, and the heating effect, caused by the jet flows in expansion space from compression space.

In a regenerator, there are two phases of gas flow, cold blow and hot blow. During the hot blow, the regen matrix collects and stores the heat from hot gas incoming from the heater. During the cold blow, the cold gas coming from the cooler collects the stored heat and warms up. The solid matrix inside the regenerator is at a slightly lower temperature than the incoming gas during the hot blow, and similarly, the solid matrix is at a slightly higher temperature than the incoming gas during the cold blow. This is to allow a temperature gradient between the solid and gas matrix and hence the heat transfer coefficient between them varies with time. When the flow reaches the end of the jetting region, it is mostly confined to the central region of the matrix. A high-temperature gradient in the axial direction identifies the jetting effects into the matrix; this is because of higher heat transfer rates at the entrance which gradually drop as it moves through the regenerator.

Figure 13 shows the indicated power of the GPU-3 engine with air as working fluid and a mean pressure of 4.14 MPa. The power generated is nearly half of that of power produced by helium under the same working conditions of temperature and pressure. The engine produces 1.64 kW at 2000 rpm as compared to 3.2 kW produced by an engine with helium as working fluid. The power produced decreases with the increase in the speed of the engine. This behavior can be attributed to lower thermal conductivity, lower specific heat, and higher density of air as compared to helium.

A β-type Stirling engine with rhombic drive, GPU-3 engine, with a mean pressure of 4.14 MPa, and air as working fluid was simulated. The following can be concluded from the study:

• Impinging effect was observed in β-type rhombic drive GPU-3 Stirling engine which led to the formation of vortices. Such vortices can be observed only by CFD analysis.

• There was a reduction in the power output of the engine of about 50% with air as a working fluid, compared to helium.

• The power output of the engine decreases with an increase in engine speed, indicating that air reduces the optimal engine speed as compared to helium (Fig. 13).

• Because of the intricate relationship between fluid dynamics and heat transfer processes inside the engine, the average gas temperature's instantaneous variation within each engine component is distant from harmonic distribution (Fig. 10).

The present study can be expanded to a Stirling engine powered by concentrated solar energy with air as working fluid, with relevant modifications to the heater section of the engine, this can contribute to the renewable energy mix. Techno-economic analysis and CFD study focusing on optimizing the engine design for this application, as well as for hybrid systems can be useful.

There are no conflicts of interest.

The authors attest that all data for this study are included in the article.

d =

diameter, m

e =

eccentricity, m

h =

heat transfer coefficient, W/m² K

k =

turbulent kinetic energy, J/kg

m =

mass, kg

p =

pressure, Pa

t =

time, s

w =

work, J/cycle

y =

vertical distance, m

A =

area, m²

K =

thermal conductivity, W/m K

L =

length, m

P =

power, W

R =

gas constant, J/kg K

T =

temperature, K

V =

volume, m³

$Q˙$ =

heat transfer rate, W

u_r =

relative velocity, m/s

A_fs =

interfacial area density, 1/m

C₂ =

inertial resistance coefficient

C_f =

friction coefficient

C_p =

specific heat

R_c =

crank radius, m

S_i =

momentum source term, N/m³

y⁺ =

dimensionless wall distance

a₁,a₂ =

friction factor correlation constants

a₃,a₄ =

Nusselt number correlation constants

Nu =

Nusselt number

Pr =

Prandtl number

Re =

Reynolds number

u, v, w =

velocity components, m/s

1/α =

viscous resistance coefficient

Α =

thermal diffusivity, m²/s

$γ$ =

regenerator porosity

ε =

dissipation rate, m²/s³

Ɵ =

crank angle, deg

λ =

bulk viscosity coefficient, Pa s

μ =

dynamic viscosity, Pa s

ρ =

density, kg/m³

ψ =

viscous dissipation, W/m³

$ωr$ =

angular frequency, rad/s

b =

bore

c =

compression

cr =

connecting rod

cs =

cross-sectional

d =

displacer

e =

expansion

eff =

effective

f =

fluid

h =

hydraulic

i, j, k =

coordinate directions in index notation

ind =

indicated

p =

piston

r =

regenerator

s =

solid

t =

turbulent

tot =

total

w =

wire