# DynamFluid: Development and Validation of a New GUI-Based CFD Tool for the Analysis of Incompressible Non-Isothermal Flows

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## Abstract

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## 1. Introduction

- Windows-based software, which can be run in any Windows Operating System: both 32- and 64-bit architectures are supported. When run in a 64-bit architecture, the software leverages on the larger word-size when performing computations.
- Graphical user interface for defining the geometric domain, physical model, boundary conditions, and displaying the results obtained in any simulation.
- Custom database for storing both the project domain definition and all the information generated during the simulation. The database provides the user with ODBC (Open DataBase Connectivity) interface for interacting with any ODBC client.
- Support for NASTRAN format file importing, which allows the user to import any geometry and physical definition in DMAP (Direct Matrix Abstraction Programming) language.
- Vtk file format capabilities, to export generated simulations into ASCII text Vtk files that can be visualized using Paraview [20].
- Basic meshing capabilities to sample the geometric domain. Two methods have been implemented: (a) structured meshing (linear, logarithmic) using both quadrangular elements (QUAD) and triangular elements (TRIA) for regular geometric domains, and (b) Delaunay–Voronoi meshing for irregular geometric domains.
- Software designed to run in a computer with a motherboard that may have one or several multi-core processors. This includes parallel computation of the finite element matrices, parallel assembly of the global matrices and parallel computation of the right hand side of each step of the algorithm. The software uses the conjugate gradient stabilized algorithm provided by the Eigen library [21] to solve the linear systems, and it has been compiled with the openmp compiler flag so that the Eigen library exploits the multiple cores available in the hardware.
- Custom user language for post-processing the results (velocity, pressure and temperature), with basic algebra functions and support for different coordinate reference systems. Internal compiler for translating this user language into machine code that can be applied in parallel to every node and/or finite element.
- Support for different types of finite elements (both Lagrangian and Serendipity): (a) linear TRIA elements and (b) linear and quadratic QUAD elements.
- DynamFluid is a freeware CFD tool available at https://sites.google.com/view/dynamfluid.

- (i)
- The lid-driven cavity flow.This is a classical benchmark problem that has been widely used since the early days of CFD to assess and validate new techniques and methods. This test case is easy to set and simulate because its boundary conditions are particularly simple. However, the fully developed flow displays almost all fluid mechanical phenomena, with increasingly complex aspects emerging as the Reynolds number is increased, such as corner eddies, laminar to turbulence regime transition, and even turbulence at high Reynolds number. A recent comprehensive review of the literature on the subject can be found in [22], where the work of several authors is presented and discussed [23,24,25,26,27,28]. Available benchmark results have been tabulated to provide a comprehensive source of validation data [29].
- (ii)
- Mixed convection flow in a vertical channel with asymmetric wall temperatures. In this mixed convection heat transfer problem, an initially uniform flow develops in a slender vertical channel whose walls are at different temperatures. The cold and hot wall temperatures may also differ from the incoming flow temperature. As a result of the upward buoyancy force that appears near the hot wall, the velocity increases in the near-wall region. As the fluid accelerates downstream, the fluid near the cold wall may suffer flow reversal so as to maintain the imposed fixed flow rate. One of the most interesting features exhibited by this flow is thus the possibility of flow reversal at the cold wall as the flow develops. The occurrence (or not) of flow reversal depends on the length of the vertical channel and the buoyancy effect induced by the temperature difference between the hot and cold walls. Previous studies have shown that for the flow reversal to occur, in high Reynolds number flows the ratio of the Grashof number to the Reynolds number must be higher than a critical value that depends on the wall temperature difference ratio [30,31,32,33,34,35,36,37,38,39,40,41,42,43].
- (iii)
- Unsteady incompressible flow past a circular cylinder. The flow of a constant density fluid past a bluff body is another classical problem that has been widely studied in the literature. Understanding the flow regimes past bluff bodies poses a daunting challenge, so that 2D and 3D vortical structures in wakes of different bodies have been analyzed by scientists and engineers for decades. The reason for this interest is the vast range of applications of external flow past round bluff bodies: aerodynamics (planes, rockets, ground vehicles), hydrodynamics (ships, submarines) or wind energy (wind turbines), to name the few. At very low Reynolds numbers (creeping flow) the flow past a non-heated circular cylinder is symmetric in the streamwise direction [44,45,46,47]. As the Reynolds number grows to values of order unity the symmetry is lost, and when it exceeds a critical value the non-linear convective effects trigger the onset of steady flow separation, accompanied by the appearance of steady recirculation bubbles behind the cylinder. One important aspect of these flows is the so-called vortex shedding, which is an oscillating flow pattern that emerges for even larger Reynolds numbers. This regime has been thoroughly studied experimentally [48,49,50,51,52] and numerically [53,54,55,56,57]. The alternate shedding of vortices in the wake leads to the well known Kármán vortex street, which originates large fluctuating pressure forces in the direction transverse to the flow and may cause structural vibrations, acoustic noise, or resonance, which in some cases may lead to structural damage or even collapse.
- (iv)
- Steady non-isothermal flow past a circular cylinder with negligible buoyancy effects. This case is similar to the previous one but with the particularity that the temperature of the cylinder differs from the temperature of the incoming fluid, which causes the flow past a circular cylinder to exhibit interesting heat transfer features. This problem is of interest for the design of cylinder-shaped sensors located in fluid streams, hot-wire anemometers, tube heat exchangers, nuclear reactor fuel rods and chimneys. In this work, attention will be restricted to steady flow with negligible buoyancy effects, with the aim of characterizing the local Nusselt number at the cylinder wall for different Reynolds numbers [44,54,58,59].

## 2. Governing Equations

## 3. Numerical Method

#### 3.1. Temporal Discretization: The Characteristics-Based-Split Algorithm

- Calculate $\mathsf{\Delta}{u}_{i}^{**}$ from Equation (19).
- Calculate $\mathsf{\Delta}\rho $ or $\mathsf{\Delta}p$ from Equation (21).
- Calculate $\mathsf{\Delta}{u}_{i}$ from Equation (20), which yields the velocity at time ${t}_{n+1}$.
- Calculate $\mathsf{\Delta}e$ from Equation (22), which can be calculated in parallel with the other steps since the right hand side of Equation (22) does not depend on the variables at time ${t}_{n+1}$. Step 4 allows obtaining the value of the energy at time ${t}_{n+1}$.

#### 3.2. Spatial Discretization

## 4. Software Validation

#### 4.1. Lid-Driven Cavity Flow

#### 4.1.1. Literature Review

#### 4.1.2. Boundary and Initial Conditions

#### 4.1.3. Convergence Analysis

#### 4.1.4. Results for $\mathrm{Re}$ up to 10,000

#### 4.2. Mixed Convection Flow in a Vertical Channel with Asymmetric Wall Temperatures

#### 4.2.1. Literature Review

#### 4.2.2. Boundary and Initial Conditions

- Left, cold non-slip wall: $u=v=\mathsf{\Theta}-{\mathsf{\Theta}}_{c}=0$.
- Right, hot non-slip wall: $u=v=\mathsf{\Theta}-1=0$.
- Bottom side, incoming flow: $u=v-1=\mathsf{\Theta}=0$.
- Top side, outgoing flow: $p=u=\partial v/\partial y=\partial \theta /\partial y=0$.

#### 4.2.3. Discussion of Results

#### 4.3. Isothermal Flow Past a Circular Cylinder

#### 4.3.1. Literature Review

#### 4.3.2. Computational Domain, Boundary and Initial Conditions and Mesh Generation

- Left boundary, uniform incoming flow: $u-1=v=0$.
- Right boundary, outflow boundary condition: $p=\partial u/\partial x=v=0$.
- Top and bottom boundaries, symmetry boundary condition: $\partial u/\partial y=v=0$.
- Cylinder wall, non-slip condition: $u=v=0$.

#### 4.3.3. Discussion of Results

#### 4.4. Flow Past a Heated Circular Cylinder with Forced Convection

#### 4.4.1. Literature Review

#### 4.4.2. Computational Domain and Boundary Conditions

#### 4.4.3. Discussion of Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a | Speed of sound |

B | Blockage ratio ($D/H$) |

${c}_{p}$ | Specific heat at constant pressure |

${c}_{v}$ | Specific heat at constant volume |

D | Cylinder diameter |

e | Internal energy per unit mass, $e={c}_{v}T$ |

${e}_{\mathrm{T}}$ | Total energy per unit mass, ${e}_{\mathrm{T}}=e+{u}_{i}{u}_{i}/2$ |

$\overline{\mathbf{E}}$ | Energy tensor containing the values of $\rho {e}_{\mathrm{T}}$ in every node of the mesh |

$E{r}_{ij}$ | Difference in the estimating function between mesh #i and mesh #j |

f | Frequency |

g | Acceleration of gravity |

Gr | Grashof number |

h | element size |

H | Characteristic height of the problem |

${I}_{ij}$ | Identity tensor |

k | Thermal conductivity |

L | Characteristic length of the problem |

${L}_{\varphi}$ | Distance from the inlet to the center of the cylinder |

${L}_{s}$ | Eddy length |

n | outward normal coordinate |

$\mathbf{N}$ | Shape functions |

Nu | Local Nusselt number |

p | Pressure |

$\overline{\mathbf{p}}$ | Pressure tensor containing the values of p in every node of the mesh |

Pr | Prandtl number, $\nu /\alpha $ |

Re | Reynolds number |

Ri | Richardson number, Gr/Re${}^{2}$ |

St | Strouhal number |

t | Time |

${\tilde{t}}_{i}$ | i-th component of the prescribed stress |

T | Temperature |

$\tilde{T}$ | Prescribed Temperature |

$\overline{\mathbf{T}}$ | Temperature tensor containing the values of T in every node of the mesh |

$\tilde{\nabla {T}_{i}}$ | i-th component of the prescribed temperature gradient |

${u}_{i}$ | i-th component of the velocity vector, ${(u,v,w)}^{T}$ |

${\overline{\mathbf{u}}}_{i}$ | Velocity tensor containing the i-th component of the velocity vector in every node of the mesh |

${\tilde{u}}_{i}$ | i-th component of the prescribed velocity |

${x}_{i}$ | i-th Cartesian coordinate, ${(x,y,z)}^{T}$ |

Greek letters | |

$\alpha $ | Thermal diffusivity, $k/\left(\rho {c}_{p}\right)$ |

$\beta $ | Thermal expansion coefficient, $-{\rho}^{-1}{(\partial \rho /\partial T)}_{p}$ |

$\mu $ | Dynamic viscosity |

$\varphi $ | Variable to approximate using the finite element method |

${\varphi}_{s}$ | Angle of detachment |

$\rho $ | Density |

$\nu $ | Kinematic viscosity, $\mu /\rho $ |

${\tau}_{ij}$ | deviatoric viscous stress tensor |

${\theta}_{1}$ | velocity relaxation factor |

${\theta}_{2}$ | pressure relaxation factor |

${\mathsf{\Theta}}_{c}$ | the wall temperature difference ratio, $({T}_{c}-{T}_{\infty})/({T}_{h}-{T}_{\infty})$ |

${\mathbb{T}}_{h}$ | unstructured triangulation composed by non-overlapping elements |

Subscripts | |

c | Cold boundary |

h | Hot boundary |

w | Wall |

∞ | Reference value |

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**Figure 1.**Schematic representation of the lid-driven cavity flow showing the coordinate system, dimensional parameters, and boundary conditions.

**Figure 2.**Convergence analysis for the horizontal velocity along the vertical mid line corresponding to $\mathrm{Re}=1000$ (

**left**) and triangular elements (TRIA) 20 × 20 element mesh used in the computations with DynamFluid (

**right**). The 50 × 50 and 100 × 100 meshes are finer meshes with the same topology.

**Figure 3.**Two-dimensional finite elements adjacent to node k used to compute the equivalent element size ${h}_{k}=\mathrm{min}(2{\mathrm{A}}_{j}/{l}_{j})$.

**Figure 5.**Schematic representation of the non-isothermal flow in a vertical channel with asymmetric wall temperatures showing the coordinate system, dimensional parameters, and boundary conditions.

**Figure 6.**Comparison between the fully developed velocity (

**left**) and temperature (

**right**) profiles as predicted by DynamFluid (solid lines) and by the fully developed theory of Aung and Worku [32] (symbols) corresponding to $\mathrm{Re}=100$, $\mathrm{Gr}=$ 25,000, and ${\mathsf{\Theta}}_{c}=\{0,0.5\}$.

**Figure 7.**Schematic representation of the flow past a circular cylinder showing the coordinate system, dimensional parameters, and boundary conditions. The size of the computational domain is determined by the parameters L, ${L}_{\varphi}$ and H.

**Figure 10.**Vorticity field of the flow past a circular cylinder for $\mathrm{Re}=100$ at four successive instants during the vortex shedding cycle.

**Figure 11.**Streamlines and temperature contours of the steady state solution for several Reynolds numbers and Ri = 0: (

**a**) Re = 10, (

**b**) Re = 15, (

**c**) Re = 20, (

**d**) Re = 25, and (

**e**) Re = 40. The color map in the left plots represents the modulus of the velocity vector.

**Figure 12.**Schematic description of the recirculation region showing the dimensionless eddy length (${L}_{s}$) and the separation angle (${\theta}_{s}$).

**Figure 13.**Variationof the local Nusselt number on the surface of the cylinder at different Reynolds number as predicted by DynamFluid (solid lines) and by previous authors.

Grid Comparison | ${\mathit{Er}}_{{\mathit{L}}_{1}}$ | ${\mathit{Er}}_{{\mathit{L}}_{\mathit{\infty}}}$ | ${\mathit{r}}_{\mathit{ij}}$ |
---|---|---|---|

$50\times 50$ vs. $20\times 20$ | $0.00176$ | $0.00748$ | $2.0$ |

$100\times 100$ vs. $50\times 50$ | $0.0138$ | $0.0969$ | $2.5$ |

$\mathit{D}/\mathit{H}$ | $\mathbf{St}$ |
---|---|

$1/16$ | $0.1792$ |

$1/32$ | $0.1703$ |

$1/64$ | $0.1650$ |

**Table 3.**Comparison between the Strouhal number for $\mathrm{Re}=\{100,200\}$ as predicted by DynamFluid and reported by previous authors.

$\mathbf{Re}$ | [56] | [77] | [86] | [85] | [54] | Present Work |
---|---|---|---|---|---|---|

100 | $0.165$ | $0.1649$ | $0.1569$ | $0.164$ | 0.164 | $0.165$ |

200 | − | $0.1958$ | $0.1957$ | − | 0.196 | $0.1954$ |

**Table 4.**Comparison between the eddy length (${L}_{s}$) and the separation angle (${\theta}_{s}$) for several Reynolds numbers as predicted by DynamFluid and reported by previous authors.

Variable | $\mathbf{Re}$ | [44] | [59] | [58] | [54] | [55] | DynamFluid |
---|---|---|---|---|---|---|---|

10 | $0.504$ | $0.498$ | $0.52$ | $0.504$ | $0.51$ | $0.512$ | |

15 | − | $1.162$ | $1.189$ | − | − | $1.227$ | |

$2{L}_{s}/D$ | 20 | $1.88$ | $1.844$ | $1.865$ | $1.86$ | $1.87$ | $1.866$ |

25 | − | − | $2.517$ | − | − | $2.548$ | |

40 | $4.69$ | $4.65$ | $4.424$ | $4.4$ | $4.59$ | $4.480$ | |

Variable | $\mathrm{Re}$ | [44] | [59] | [58] | [54] | [55] | DynamFluid |

10 | $29.6$ | $29.3$ | $29.12$ | $30.0$ | − | $28.57$ | |

15 | − | $38.6$ | $38.57$ | − | − | $38.57$ | |

${\theta}_{s}$ (${}^{\circ}$) | 20 | $43.7$ | $43.65$ | $43.64$ | $44.1$ | − | $43.58$ |

25 | − | − | $46.89$ | − | − | $47.14$ | |

40 | $53.8$ | $53.55$ | $53.1$ | $53.5$ | − | $51.43$ |

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## Share and Cite

**MDPI and ACS Style**

Redal, H.; Carpio, J.; García-Salaberri, P.A.; Vera, M. DynamFluid: Development and Validation of a New GUI-Based CFD Tool for the Analysis of Incompressible Non-Isothermal Flows. *Processes* **2019**, *7*, 777.
https://doi.org/10.3390/pr7110777

**AMA Style**

Redal H, Carpio J, García-Salaberri PA, Vera M. DynamFluid: Development and Validation of a New GUI-Based CFD Tool for the Analysis of Incompressible Non-Isothermal Flows. *Processes*. 2019; 7(11):777.
https://doi.org/10.3390/pr7110777

**Chicago/Turabian Style**

Redal, Héctor, Jaime Carpio, Pablo A. García-Salaberri, and Marcos Vera. 2019. "DynamFluid: Development and Validation of a New GUI-Based CFD Tool for the Analysis of Incompressible Non-Isothermal Flows" *Processes* 7, no. 11: 777.
https://doi.org/10.3390/pr7110777