I want to plot the legend of my fill aesthetic at the bottom and I want a) the legend to be in the middle of the plot (right under the x-axis label) and b) I want the title of the legend to be aligned to the centre of the legend.

Code to create the figure above

# Libraries
library(ggplot2)
library(plyr)
library(latex2exp)

# Function to calculate
calc_wet_bulb_temperature <- function(dry_temp, RH){
  wet_temp <- dry_temp*atan(0.151977 * sqrt(RH + 8.313659)) +
                  0.00391838 * sqrt(RH^3) * atan(0.023101*RH) -
                  atan(RH - 1.676331) + atan(dry_temp + RH) - 4.686035
  return(wet_temp)
}

# Space to visualise
dry_temp   <- seq(from = -10, to = 45, by = 1)
RH         <- seq(from = 0, to = 100, by = 1)
temp_space <- expand.grid(dry_temp, RH)
names(temp_space) <- c("dry_temp", "RH")

# Calculate wet bulb temperature
temp_space <- ddply(temp_space, c("dry_temp", "RH"), summarise,
                    wet_temp = calc_wet_bulb_temperature(dry_temp, RH))

# Visualise space
## Calculate value where the colour scale should be red
#diff(range(temp_space$wet_temp))
#diff(c(range(temp_space$wet_temp)[1], 30))
#42.69626/57.88312
## Actual plot
ggplot(temp_space, aes(x = RH, y = dry_temp, fill = wet_temp)) +
  geom_tile() +
  coord_equal(expand = FALSE) +
  scale_fill_viridis_c(option = "H", 
                       values = c(0.0, 0.2, 0.4, 0.6, 0.7376289, 1.0),
                       limits = range(temp_space$wet_temp)) +
 theme(legend.position = "bottom", 
       legend.title.position = "top") +
  labs(title = "Wet - bulb temperature", 
       x = "Relative humidity", 
       y = TeX(r"( Dry temperature ($C\degree$) )"),
       fill = TeX(r"( Wet temperature ($C\degree$) )"))
  

Add plot.title=element_text(hjust = 0.5) and legend.key.size = unit(0.8, "cm") to your theme.

  theme(legend.position = "bottom", 
        legend.title.position = "top",
        plot.title = element_text(hjust = 0.5),
        legend.key.size = unit(0.8, "cm"))